GCSE Maths Angles, Polygons and Symmetry (Foundation): worked examples and questions with answers
By the end of this chapter you will be able to
- Use the correct words and notation for points, lines, angles, polygons and triangles
- Use angles on a line, angles around a point and vertically opposite angles, and give reasons
- Find missing angles in triangles, including isosceles and equilateral triangles
- Use alternate, corresponding and co-interior angles on parallel lines
- Know the properties of the special quadrilaterals and use their angle sums
- Calculate interior and exterior angles of polygons and regular polygons
- Find lines of symmetry and the order of rotational symmetry
- Solve angle problems with algebra and write clear reasons (Higher: geometric proof)
You should already know
- Measuring and drawing angles with a protractor
- Solving simple linear equations (Chapter 9)
- Basic names of 2D shapes
Where this leads
- Pythagoras and trigonometry (Chapter 20)
- Congruence and similarity (Chapter 21)
- Circle theorems (Chapter 26)
Edexcel GCSE (9–1) Mathematics 1MA1, Foundation tier. Content references: G1, G3, G4, G6.
- 17.1 Geometry words and notation: Worksheet (PDF) · Answers (PDF) · lesson page
- 17.2 Angles on a line, around a point and vertically opposite: Worksheet (PDF) · Answers (PDF) · lesson page
- 17.3 Angles in triangles: Worksheet (PDF) · Answers (PDF) · lesson page
- 17.4 Parallel lines: Worksheet (PDF) · Answers (PDF) · lesson page
- 17.5 Quadrilaterals: Worksheet (PDF) · Answers (PDF) · lesson page
- 17.6 Interior and exterior angles of polygons: Worksheet (PDF) · Answers (PDF) · lesson page
Exercise P Do you remember?
Non-calculatorMethod. m
- a bc
- a bc
Geometry words and notation
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Geometry is the maths of shapes, lines and angles. To talk about shapes without confusion, mathematicians use a small set of exact words. This section teaches the words, so that the rest of the chapter makes sense.
A point marks an exact position. It has no size, and we usually name it with a capital letter such as A. A line is straight and goes on for ever in both directions. A line segment is the piece of a line between two points. The line segment from A to B is written AB. A plane is a flat surface that goes on for ever, like an enormous sheet of paper; all the 2D shapes in this chapter are drawn on a plane.
Two lines are parallel if they always stay the same distance apart and never meet, like the two rails of a straight railway track. We show this with matching arrows. Two lines are perpendicular if they meet at a right angle (90°), like the corner of a page. We show this with a small square.
An angle measures a turn. We measure it in degrees. A full turn is 360°. Angles have names that depend on their size.
Naming an angle. We use three letters. The angle ABC is the angle at the middle letter, B, between the lines BA and BC. The vertex is always the middle letter. If a diagram only has one angle at a corner, you can sometimes use one letter, but three letters are always safe.
A polygon is a closed 2D shape with straight sides. The corners are called vertices (one corner is a vertex) and the sides are called edges. A polygon with all sides equal and all angles equal is a regular polygon. The names come from the number of sides: 3 triangle, 4 quadrilateral, 5 pentagon, 6 hexagon, 7 heptagon, 8 octagon, 9 nonagon, 10 decagon.
Triangles. By convention we call the vertices A, B, C and the sides opposite them a, b, c. So side a is opposite angle A (it does not touch A). The side from A to B is called AB, and it is opposite C.
Diagrams use marks to give information. Small dashes on two sides (ticks) mean those sides are equal in length. Arrows mean parallel. A small square means 90°. Matching arcs mean equal angles. Diagrams are often “not drawn accurately”, so do not measure; use the marks.
Key ideas
- A point is a position; a line goes on for ever; a line segment AB joins two points.
- Parallel lines never meet (arrows). Perpendicular lines meet at 90° (small square).
- Acute < 90°; right angle = 90°; obtuse 90° to 180°; straight line = 180°; reflex 180° to 360°; full turn = 360°.
- Angle ABC is the angle at B, the middle letter, between BA and BC.
- A polygon has straight sides; its corners are vertices and its sides are edges. A regular polygon has equal sides and equal angles.
- In triangle ABC, side a is opposite angle A, side b is opposite B, and side c is opposite C.
Key words. The vertex is the corner. Opposite means on the other side of the shape from the vertex.
Method. The angle is named with three letters, with the vertex in the middle. The side opposite an angle is the side that does not touch it.
- Find the vertex of the angle. The vertex is the corner where the angle is. Here the angle is marked at Q.Vertex = Q
- Write the other two letters on either side. The two arms of the angle go from Q to P and from Q to R, so the letters P and R go at the ends and Q goes in the middle.Angle PQR (or angle RQP)
- Find the opposite side. The opposite side joins the two other vertices, P and R, so it does not touch Q.The side opposite Q is PR.
Check. The side PR does not contain the letter Q, so it cannot touch the vertex Q. ✓
Watch out. The vertex letter must be in the middle: PQR, not QPR (QPR would be the angle at P).
Key words. Acute, obtuse and reflex describe the size of an angle.
Method. Compare each angle with the benchmarks 90°, 180° and 360°.
- List the benchmarks. Each name belongs to a range of sizes.acute: less than 90°. right: 90°. obtuse: 90° to 180°. reflex: 180° to 360°.
- Name (a) and (b). 64 is less than 90, so it is small and sharp. 90 is exactly a quarter turn.(a) 64°: acute
(b) 90°: right angle - Name (c) and (d). 138 is bigger than 90 but smaller than 180. 215 is bigger than 180 but smaller than 360.(c) 138°: obtuse
(d) 215°: reflex
Check. Order them: 64° < 90° < 138° < 215° goes acute, right, obtuse, reflex. ✓
Watch out. A straight line is exactly 180°; it is neither obtuse nor reflex.
Key words. A sketch is a rough drawing. An isosceles triangle has two equal sides.
Method. Draw a sketch from the words: read each fact, draw it, and mark it on the diagram.
- Draw the line segment AB. We know one side is 6 cm, so we start by drawing it and labelling its ends.A ---------- B (6 cm)
- Draw the angle at A. Angle BAC means the angle at the middle letter A between AB and AC. We draw 40° at A.angle BAC = 40°
- Make AC the same length as AB. The question says AB = AC, so side AC is also 6 cm. We join C to B to finish the triangle and mark the equal sides with matching ticks.AB = AC = 6 cm (one tick on each)
- Name the triangle. A triangle with two equal sides is isosceles.Isosceles triangle
Check. Two sides have ticks, so two sides are equal, which is the definition of isosceles. ✓
Watch out. A sketch does not have to be exact, but it must show every fact that the question gives (lengths, angles, equal sides).
Watch out
Angle ABC is the angle at B, the middle letter. Students often put the vertex letter first or last. And remember: parallel lines never meet, perpendicular lines meet at a right angle. They are not the same thing.Exercise 17A Words, notation and naming angles
Non-calculatorKey words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.parallel: lines that stay the same distance apart and never meet
- Match the description to the word. The two lines never meet and stay the same distance apart.The answer is parallel.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.perpendicular: lines that meet at a right angle (90°)
- Match the description to the word. The lines meet at 90°.The answer is perpendicular.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.vertex: a corner of a shape (the plural is vertices)
- Match the description to the word. It is a corner.The answer is vertex.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.polygon: a closed 2D shape with straight sides
- Match the description to the word. It is flat, closed and has straight sides.The answer is polygon.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.regular polygon: a polygon with equal sides and equal angles
- Match the description to the word. All sides and all angles are equal.The answer is regular polygon.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.line segment: the part of a line between two points
- Match the description to the word. It has two end points.The answer is line segment.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. An acute angle is small and sharp. An obtuse angle is wide. A reflex angle is more than half a turn.
Method. Compare the size of the angle with 90°, 180° and 360°.
- Remember the sizes of the angle names. Acute is less than 90°, a right angle is 90°, obtuse is between 90° and 180°, a straight line is 180°, reflex is between 180° and 360°.acute < 90° < obtuse < 180° < reflex < 360°
- Place 35° on that list. Look at which two benchmarks it is between.35° is between 0° and 90°.
- Write the name. acute
Check. Picture the angle as a turn: a quarter turn is 90°, a half turn is 180°. Does your answer fit the picture? ✓
Watch out. An obtuse angle is bigger than 90° but smaller than 180°. A reflex angle is bigger than 180°.
Key words. An acute angle is small and sharp. An obtuse angle is wide. A reflex angle is more than half a turn.
Method. Compare the size of the angle with 90°, 180° and 360°.
- Remember the sizes of the angle names. Acute is less than 90°, a right angle is 90°, obtuse is between 90° and 180°, a straight line is 180°, reflex is between 180° and 360°.acute < 90° < obtuse < 180° < reflex < 360°
- Place 90° on that list. Look at which two benchmarks it is between.90° is exactly 90°.
- Write the name. right angle
Check. Picture the angle as a turn: a quarter turn is 90°, a half turn is 180°. Does your answer fit the picture? ✓
Watch out. An obtuse angle is bigger than 90° but smaller than 180°. A reflex angle is bigger than 180°.
Key words. An acute angle is small and sharp. An obtuse angle is wide. A reflex angle is more than half a turn.
Method. Compare the size of the angle with 90°, 180° and 360°.
- Remember the sizes of the angle names. Acute is less than 90°, a right angle is 90°, obtuse is between 90° and 180°, a straight line is 180°, reflex is between 180° and 360°.acute < 90° < obtuse < 180° < reflex < 360°
- Place 150° on that list. Look at which two benchmarks it is between.150° is between 90° and 180°.
- Write the name. obtuse
Check. Picture the angle as a turn: a quarter turn is 90°, a half turn is 180°. Does your answer fit the picture? ✓
Watch out. An obtuse angle is bigger than 90° but smaller than 180°. A reflex angle is bigger than 180°.
Key words. An acute angle is small and sharp. An obtuse angle is wide. A reflex angle is more than half a turn.
Method. Compare the size of the angle with 90°, 180° and 360°.
- Remember the sizes of the angle names. Acute is less than 90°, a right angle is 90°, obtuse is between 90° and 180°, a straight line is 180°, reflex is between 180° and 360°.acute < 90° < obtuse < 180° < reflex < 360°
- Place 180° on that list. Look at which two benchmarks it is between.180° is exactly 180°.
- Write the name. straight line
Check. Picture the angle as a turn: a quarter turn is 90°, a half turn is 180°. Does your answer fit the picture? ✓
Watch out. An obtuse angle is bigger than 90° but smaller than 180°. A reflex angle is bigger than 180°.
Key words. An acute angle is small and sharp. An obtuse angle is wide. A reflex angle is more than half a turn.
Method. Compare the size of the angle with 90°, 180° and 360°.
- Remember the sizes of the angle names. Acute is less than 90°, a right angle is 90°, obtuse is between 90° and 180°, a straight line is 180°, reflex is between 180° and 360°.acute < 90° < obtuse < 180° < reflex < 360°
- Place 270° on that list. Look at which two benchmarks it is between.270° is between 180° and 360°.
- Write the name. reflex
Check. Picture the angle as a turn: a quarter turn is 90°, a half turn is 180°. Does your answer fit the picture? ✓
Watch out. An obtuse angle is bigger than 90° but smaller than 180°. A reflex angle is bigger than 180°.
Key words. An acute angle is small and sharp. An obtuse angle is wide. A reflex angle is more than half a turn.
Method. Compare the size of the angle with 90°, 180° and 360°.
- Remember the sizes of the angle names. Acute is less than 90°, a right angle is 90°, obtuse is between 90° and 180°, a straight line is 180°, reflex is between 180° and 360°.acute < 90° < obtuse < 180° < reflex < 360°
- Place 89° on that list. Look at which two benchmarks it is between.89° is between 0° and 90°.
- Write the name. acute
Check. Picture the angle as a turn: a quarter turn is 90°, a half turn is 180°. Does your answer fit the picture? ✓
Watch out. An obtuse angle is bigger than 90° but smaller than 180°. A reflex angle is bigger than 180°.
Method. Name an angle with three letters. The vertex (the corner where the arc is) goes in the middle.
- Find the corner where the arc is drawn. The arc shows which angle is meant. It is drawn at the vertex.The arc is at Q.
- Put Q in the middle. The middle letter of the name is always the vertex.? Q ?
- Add the letters at the ends of the two arms. The arms go from Q to P and from Q to R.angle PQR (or angle RQP)
Check. The middle letter is Q, the corner with the arc. ✓
Watch out. Writing PRQ would be the angle at R.
Method. The side opposite a vertex is the side that does not touch it.
- Find the vertex. We are looking at A.Vertex A
- Look for the side that does not touch A. The sides at A are AB and AC, so the other side, BC, is opposite.BC, which is called a.
Check. Side a is opposite A, b is opposite B, c is opposite C. ✓
Method. The side opposite a vertex is the side that does not touch it.
- Find the vertex. We are looking at B.Vertex B
- Look for the side that does not touch B. The sides at B are AB and BC, so the other side, AC, is opposite.AC, which is called b.
Check. Side b has no letter B in AC. ✓
Method. Opposite pairs go together: a with A, b with B, c with C.
- Recall the pairing. Lower case letters name the sides opposite the capital letters.a ↔ A, b ↔ B, c ↔ C
- Pick side c. The opposite vertex is C.angle C
Check. The side c is AB, and AB does not touch C. ✓
Method. Look at the arrows and the small squares on the diagram.
- Look for arrows. Arrows on two lines show they are parallel.AB and DC both have an arrow.
- Write the word. Lines with matching arrows are parallel.parallel
Check. The opposite sides of a rectangle are parallel. ✓
Method. Look at the small square at B.
- Look for the small square. A small square means 90°.There is a small square at B, between AB and BC.
- Write the word. Lines meeting at 90° are perpendicular.perpendicular
Check. Adjacent sides of a rectangle meet at a right angle. ✓
Method. An edge joins two neighbouring vertices.
- Recall the meaning. An edge is a side of the shape; a vertex is a corner.edge = side, vertex = corner
- Choose a side. Side AB has the corners A and B at its ends.A and B (any neighbouring pair is correct)
Check. AB is a side of the rectangle, so A and B are joined by an edge. ✓
Method. Draw each fact on the diagram one at a time.
- Draw two lines meeting at B. Angle ABC has its vertex at B, so B is the corner of the angle. It is 90°, so the two lines are perpendicular, and we mark a small square.BA and BC at right angles
- Make the two sides the same length. AB = BC means the two sides at the right angle are equal. We draw them equal and mark one tick on each.AB = BC
- Join A to C and name the triangle. Two equal sides make it isosceles, and it has a right angle, so it is right-angled.isosceles right-angled triangle
Check. Both marks are on the diagram: the right-angle square at B and the ticks on AB and BC. ✓
Watch out. Marks matter. A sketch with no ticks and no right-angle square does not show all the information.
Method. A polygon has the same number of sides as vertices.
- Recall the name. The name of a polygon tells us the number of sides.pentagon = 5 sides
- Count the corners. Each side joins two corners, and every corner is joined by two sides, so there are the same number of each.5 sides and 5 vertices
Check. Sides and vertices are always equal for a polygon. ✓
Method. A polygon has the same number of sides as vertices.
- Recall the name. The name of a polygon tells us the number of sides.hexagon = 6 sides
- Count the corners. Each side joins two corners, and every corner is joined by two sides, so there are the same number of each.6 sides and 6 vertices
Check. Sides and vertices are always equal for a polygon. ✓
Method. A polygon has the same number of sides as vertices.
- Recall the name. The name of a polygon tells us the number of sides.octagon = 8 sides
- Count the corners. Each side joins two corners, and every corner is joined by two sides, so there are the same number of each.8 sides and 8 vertices
Check. Sides and vertices are always equal for a polygon. ✓
Method. A polygon has the same number of sides as vertices.
- Recall the name. The name of a polygon tells us the number of sides.decagon = 10 sides
- Count the corners. Each side joins two corners, and every corner is joined by two sides, so there are the same number of each.10 sides and 10 vertices
Check. Sides and vertices are always equal for a polygon. ✓
Method. A cube has a vertex at every corner.
- Count the corners of the top face. The top face is a square with 4 corners.4 vertices on top
- Count the corners of the bottom face. The bottom face has 4 corners as well.4 vertices underneath
- Add. Every vertex of a cube is on the top or the bottom.4 + 4 = 8
Check. A cube has 8 vertices and 12 edges; 8 − 12 + 6 = 2 (Euler's formula). ✓
Method. An edge is where two faces meet.
- Count the edges around the top face. A square has 4 sides.4 edges on top
- Count the edges around the bottom face and the upright edges. The bottom square has 4 edges and 4 edges join the top to the bottom.4 + 4 = 8, then 8 + 4 = 12
Check. Vertices − edges + faces = 8 − 12 + 6 = 2. ✓
Method. Check the definition: a rectangle is a quadrilateral with four right angles.
- Recall the definition of a rectangle. A rectangle has four right angles and opposite sides equal.rectangle: 4 right angles
- Check a square against it. A square has four right angles, so it meets the definition.A square is a special rectangle with all sides equal.
Check. Every square is a rectangle, but not every rectangle is a square. ✓
Watch out. A shape can belong to more than one family.
Method. Compare the definitions of parallel and perpendicular.
- Recall parallel. Parallel lines never meet.parallel = never meet
- Recall perpendicular. It is perpendicular lines that meet at 90°.perpendicular = meet at 90°
- Decide. The statement mixes the two words.False
Check. Railway tracks are parallel and they never meet. ✓
Method. Regular means all sides equal and all angles equal.
- Recall hexagon. A hexagon has 6 sides.hex = 6
- Recall regular. Regular means all sides and all angles are equal.equal sides and equal angles
- Decide. Both parts of the statement match the definitions.True
Check. The words match the definitions exactly. ✓
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.equilateral: a triangle with three equal sides (and three equal angles)
- Match the description to the word. All three sides are equal.The answer is equilateral.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.isosceles: a triangle with two equal sides (and two equal angles)
- Match the description to the word. Two sides are equal.The answer is isosceles.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.scalene: a triangle with three different side lengths
- Match the description to the word. No sides are equal.The answer is scalene.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.right-angled: a triangle that has one angle of 90°
- Match the description to the word. There is a 90° angle.The answer is right-angled.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Method. Draw the three lines and look at how PQ and RS are positioned compared with each other.
- Draw the line TU. This is the line that both lines are perpendicular to.a horizontal line TU
- Draw PQ and RS at right angles to TU. Perpendicular means they cross TU at 90°. Two lines both standing straight up from the same horizontal line point in the same direction.PQ and RS are both vertical
- Decide. Two lines in the same direction never meet and stay the same distance apart.PQ and RS are parallel
Check. The railway sleepers are perpendicular to both rails, and the rails are parallel. ✓
Watch out. This is true in a flat plane (a flat sheet).
Method. Draw each fact from the question one at a time.
- Draw PQ. Start with one side and name its ends.PQ drawn horizontally
- Draw QR at 90°. Angle PQR is the angle at Q between QP and QR. It is a right angle, so QR is perpendicular to PQ.QR vertical, small square at Q
- Draw SR parallel to PQ. Parallel lines go the same way, so SR is horizontal, and we put matching arrows on PQ and SR.arrows on PQ and SR
- Join S to P. This finishes the quadrilateral.PQRS is a right-angled trapezium (or a rectangle)
Check. Every fact in the question appears as a mark on the sketch. ✓
Watch out. Check the middle letter when drawing an angle: PQR is at Q.
Method. The letters name two points, and AB is the line segment between them.
- Read the letters. AB means the segment from A to B.A to B
- Add the length. = 5 cm gives its length.AB is 5 cm long.
Check. AB and BA are the same segment. ✓
Method. Three letters name an angle and the middle letter is the vertex.
- Find the vertex. The middle letter is Y.vertex Y
- Say the size. The angle between YX and YZ is 40°.angle XYZ is 40°.
Check. The vertex letter Y is in the middle. ✓
Angles on a line, around a point and vertically opposite
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Three angle facts are the foundation of everything in this chapter. You will use them in almost every question, so learn them well.
Angles on a straight line add up to 180°. A straight line is a half turn, and a half turn is 180°. If two or more angles sit side by side on a straight line, their sizes make 180° altogether. For example, if one angle on a line is 112°, the angle next to it must be 180° − 112° = 68°.
Angles around a point add up to 360°. A full turn is 360°. If angles meet at one point and fill all the space around it, they make a full turn, so they add up to 360°.
Vertically opposite angles are equal. When two straight lines cross, they make an X. The angles that face each other across the crossing point are called vertically opposite angles. (“Vertically” here has nothing to do with up and down; it refers to the vertex, the point where the lines cross.) Vertically opposite angles are always equal. Why? Look at the picture: angle a and angle b are both on a straight line with the same angle c, so a = 180° − c and b = 180° − c. They must be equal.
Giving reasons. In an exam you are often told to “give a reason”. A good reason names the angle fact in words, exactly as above. Write the reason next to each step: “x = 68° (angles on a straight line add up to 180°)”. Never write “because it is the same” or “because it looks like it”.
Key ideas
- Angles on a straight line add up to 180°.
- Angles around a point add up to 360°.
- Vertically opposite angles are equal.
- Method: (1) decide which fact fits, (2) add the angles you know, (3) subtract from the total, (4) write the reason.
- Angles on a line are next to each other; vertically opposite angles are opposite each other at an X.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.112 = 112°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 112 = 68°
x = 68°
Check. Add all the angles: 112 + 68 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles around a point must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A full turn is 360°. Angles that meet at a point and fill the space around it share one full turn.Angles around a point add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.95 + 120 + 70 = 285°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 285 = 75°
x = 75°
Check. Add all the angles: 95 + 120 + 70 + 75 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Use angles on a straight line first, then vertically opposite angles.
- Find a: it is next to the 64° angle on a straight line. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
a = 180 − 64 = 116° - Find b: it is opposite the 64° angle. The two angles face each other across the crossing point, so they are vertically opposite.Vertically opposite angles are equal.
b = 64° - Find c: it is opposite a. Again an X shape: c and a are vertically opposite.Vertically opposite angles are equal.
c = a = 116°
Check. Add all four angles around the crossing point: 64 + 116 + 64 + 116 = 360°. ✓
Watch out. The two 64° angles are equal, and the two 116° angles are equal. Opposite angles match; neighbouring angles add to 180°.
Watch out
Check which total to use. A straight line is 180°; a full turn around a point is 360°. If the angles are on a line and you subtract from 360°, you will get a wrong (and usually too large) answer. Also remember to write the reason in words.Exercise 17B Angles on a line, around a point and vertically opposite
Non-calculatorKey words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.112 = 112°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 112 = 68°
x = 68°
Check. Add all the angles: 112 + 68 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.48 = 48°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 48 = 132°
x = 132°
Check. Add all the angles: 48 + 132 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.35 + 90 = 125°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 125 = 55°
x = 55°
Check. Add all the angles: 35 + 90 + 55 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.70 + 40 + 55 = 165°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 165 = 15°
y = 15°
Check. Add all the angles: 70 + 40 + 55 + 15 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles around a point must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A full turn is 360°. Angles that meet at a point and fill the space around it share one full turn.Angles around a point add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.130 + 120 = 250°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 250 = 110°
a = 110°
Check. Add all the angles: 130 + 120 + 110 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles around a point must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A full turn is 360°. Angles that meet at a point and fill the space around it share one full turn.Angles around a point add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.85 + 95 + 110 = 290°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 290 = 70°
b = 70°
Check. Add all the angles: 85 + 95 + 110 + 70 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles around a point must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A full turn is 360°. Angles that meet at a point and fill the space around it share one full turn.Angles around a point add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.72 + 100 + 64 + 58 = 294°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 294 = 66°
c = 66°
Check. Add all the angles: 72 + 100 + 64 + 58 + 66 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.64 = 64°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 64 = 116°
a = 116°
Check. Add all the angles: 64 + 116 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Vertically opposite angles are equal: they are opposite each other where two straight lines cross.
- Say which angle is equal to b. The two lines cross at a point. The angles that face each other across the crossing point are vertically opposite (opposite each other at the X).The 64° angle and b are vertically opposite.
- State the angle fact. This is the reason we give in an exam.Vertically opposite angles are equal.
- Write the answer. The two angles are equal, so they have the same size.b = 64°
Check. Check: b and the 64° angle are opposite each other at the X, so they must look the same size. ✓
Watch out. Do not confuse vertically opposite angles (opposite at the X) with angles on a line (next to each other, adding to 180°).
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Angle c is opposite angle a, which we found in part (a).
- Say which angle is equal to c. The two lines cross at a point. The angles that face each other across the crossing point are vertically opposite (opposite each other at the X).Angle a (116°) and c are vertically opposite.
- State the angle fact. This is the reason we give in an exam.Vertically opposite angles are equal.
- Write the answer. The two angles are equal, so they have the same size.c = 116°
Check. Check with a different fact: c and the 64° angle are on a straight line, so c = 180 − 64 = 116°. ✓
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.127 = 127°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 127 = 53°
a = 53°
Check. Add all the angles: 127 + 53 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Vertically opposite angles are equal: they are opposite each other where two straight lines cross.
- Say which angle is equal to b. The two lines cross at a point. The angles that face each other across the crossing point are vertically opposite (opposite each other at the X).The 127° angle and b are vertically opposite.
- State the angle fact. This is the reason we give in an exam.Vertically opposite angles are equal.
- Write the answer. The two angles are equal, so they have the same size.b = 127°
Check. Check: b and the 127° angle are opposite each other at the X, so they must look the same size. ✓
Watch out. Do not confuse vertically opposite angles (opposite at the X) with angles on a line (next to each other, adding to 180°).
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Angle c is opposite angle a, which we found in part (a).
- Say which angle is equal to c. The two lines cross at a point. The angles that face each other across the crossing point are vertically opposite (opposite each other at the X).Angle a (53°) and c are vertically opposite.
- State the angle fact. This is the reason we give in an exam.Vertically opposite angles are equal.
- Write the answer. The two angles are equal, so they have the same size.c = 53°
Check. Check with a different fact: c and the 127° angle are on a straight line, so c = 180 − 127 = 53°. ✓
Method. Recall the fact.
- Think of a straight line as a half turn. A full turn is 360°, and half of 360 is 180.360 ÷ 2 = 180
- Write the answer. 180
Check. A straight line is a half turn: 180°. ✓
Method. Recall the fact.
- Think of a full turn. Going all the way round a point is one full turn.one full turn = 360°
- Write the answer. 360
Check. A full turn is 360°. ✓
Method. Recall the fact.
- Picture two lines crossing in an X. The angles that face each other across the crossing point are vertically opposite.top and bottom are a pair; left and right are a pair
- Write the answer. Each pair has the same size.equal
Check. In an X, opposite angles match. ✓
Method. Angles on a straight line must add up to 180°. Add them and compare.
- Recall the fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add the two angles. If they are on a straight line the total must be exactly 180.107 + 83 = 190°
- Compare with 180°. 190 is not 180, so the angles cannot make a straight line.No: 190° ≠ 180°
Check. If the angles were 107° and 73° they would total 180°. ✓
Watch out. A reason must mention the total: “they do not add up to 180°”.
Method. The 12 hours share a full turn of 360°.
- Find the angle for one hour. A full turn is 360° and there are 12 equal gaps.360 ÷ 12 = 30°
- Count the gaps between the hands. At 3 o’clock the hands point at 12 and at 3: 3 gaps.3 × 30 = 90°
Check. 3 o’clock is a quarter turn: 360 ÷ 4 = 90°. ✓
Method. The 12 hours share a full turn of 360°.
- Find the angle for one hour. A full turn is 360° and there are 12 equal gaps.360 ÷ 12 = 30°
- Count the gaps between the hands. At 5 o’clock the hands point at 12 and at 5: 5 gaps.5 × 30 = 150°
Check. 6 gaps would be 180°, a straight line, and 150° is a bit less. ✓
Method. Use the smaller of the two angles between the hands.
- Find the angle for one hour. A full turn is 360° and there are 12 equal gaps.360 ÷ 12 = 30°
- Count the gaps the long way. From 12 round to 8 is 8 gaps.8 × 30 = 240° (a reflex angle)
- Find the smaller angle. The two angles around the point add up to 360°.360 − 240 = 120°
Check. The short way round from 8 to 12 is 4 gaps: 4 × 30 = 120°. ✓
Watch out. Give the smaller angle unless the question says otherwise.
Method. Find the angle of one slice, then find the angle for the slices that are left.
- Find the angle of one slice. The whole pizza is a full turn, 360°, shared equally by 8 slices.360 ÷ 8 = 45°
- Find the number of slices left. Anna takes 3 of the 8.8 − 3 = 5 slices
- Find the angle for 5 slices. Each slice is 45°.5 × 45 = 225°
Check. Anna’s slices take 3 × 45 = 135°, and 135 + 225 = 360°. ✓
Watch out. Check that your angles add up to 360° round the centre.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 180°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Write an equation. The angles must add up to 180°, so we add all the expressions and put the answer equal to 180.x° + 2x° = 180°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.3x = 180
- Divide both sides by 3. 3x means 3 × x, so dividing by 3 leaves x on its own.x = 180 ÷ 3 = 60
- Substitute x = 60 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.x° = 1 × 60 = 60°
2x° = 2 × 60 = 120°
Check. Add the angles: 60 + 120 = 180°, which is 180°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 180°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Write an equation. The angles must add up to 180°, so we add all the expressions and put the answer equal to 180.(3x + 20°) + (2x + 10°) = 180°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.5x + 30 = 180
- Get the number on the other side. Subtract 30 from both sides.5x = 150
- Divide both sides by 5. 5x means 5 × x, so dividing by 5 leaves x on its own.x = 150 ÷ 5 = 30
- Substitute x = 30 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.3x + 20° = 3 × 30 + 20 = 110°
2x + 10° = 2 × 30 + 10 = 70°
Check. Add the angles: 110 + 70 = 180°, which is 180°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 360°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A full turn is 360°. Angles that meet at a point and fill the space around it share one full turn.Angles around a point add up to 360°.
- Write an equation. The angles must add up to 360°, so we add all the expressions and put the answer equal to 360.2x° + (x + 30°) + (3x − 30°) = 360°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.6x = 360
- Divide both sides by 6. 6x means 6 × x, so dividing by 6 leaves x on its own.x = 360 ÷ 6 = 60
- Substitute x = 60 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.2x° = 2 × 60 = 120°
x + 30° = 1 × 60 + 30 = 90°
3x − 30° = 3 × 60 − 30 = 150°
Check. Add the angles: 120 + 90 + 150 = 360°, which is 360°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Method. Check which fact applies to angles on a straight line.
- Recall the fact for a straight line. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Spot the mistake. Dan used 360°, which is the total for a full turn around a point, not for a straight line.wrong total: 360 instead of 180
- Do the correct calculation. Subtract from 180°.180 − 112 = 68°
Check. 112 + 68 = 180. ✓
Watch out. Ask yourself: is it a straight line (180°) or a full turn (360°)?
Method. Angles AEC and AED are next to each other on the straight line CD, so they add up to 180°.
- Say how the angles are related. Angles on a straight line add up to 180°.angle AEC + angle AED = 180°
- Use one unit for the smaller angle. If AED is 1 part, AEC is 3 parts, so there are 4 parts altogether.1 + 3 = 4 parts = 180°
- Find one part. Divide by 4.180 ÷ 4 = 45°
Check. AEC = 3 × 45 = 135 and 135 + 45 = 180°. ✓
Method. Look at which angle is opposite angle BEC where AB and CD cross.
- Spot the pair. Angle BEC is opposite angle AED because AB and CD are straight lines crossing at E.BEC and AED are vertically opposite
- Use the fact. Vertically opposite angles are equal.angle BEC = angle AED = 45°
Check. BEC and AEC are on a straight line: 45 + 135 = 180°. ✓
Angles in triangles
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The three angles of any triangle add up to 180°. You can see why with a piece of paper: draw any triangle, tear off its three corners, and place them side by side with the points touching. They always fit together to make a straight line, and a straight line is 180°. (Section 17.9 explains a proper proof.)
To find a missing angle, add the two you know and subtract the total from 180°.
Isosceles triangles. An isosceles triangle has two equal sides. The angles opposite the equal sides are also equal. These are called the base angles. The ticks on the diagram show which sides are equal. If you know the angle at the top (the apex), subtract it from 180° and share the rest equally between the two base angles. If you know a base angle, the other base angle is the same, and the apex is 180° minus the two base angles.
Equilateral triangles. An equilateral triangle has three equal sides and three equal angles. Each angle is 180° ÷ 3 = 60°.
Right-angled triangles have one angle of 90°, so the other two angles add up to 90°. A triangle with all angles less than 90° is acute-angled, and one with an angle bigger than 90° is obtuse-angled. A scalene triangle has no equal sides.
Exterior angles. If you extend one side of a triangle, you make an exterior angle outside the triangle. The exterior angle and the interior angle next to it are on a straight line, so they add up to 180°. This leads to a very useful shortcut: an exterior angle of a triangle equals the sum of the two opposite interior angles (the two corners it is not next to).
Key ideas
- Angles in a triangle add up to 180°.
- Isosceles: two equal sides, and the base angles (opposite the equal sides) are equal.
- Equilateral: three equal sides and three angles of 60°.
- Right-angled: one angle is 90°, so the other two add to 90°.
- Exterior angle of a triangle = the sum of the two opposite interior angles.
- Exterior angle + its interior angle = 180° (straight line).
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the triangle must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.52 + 71 = 123°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 123 = 57°
x = 57°
Check. Add all the angles: 52 + 71 + 57 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Take the apex angle away from 180° and share what is left between the two base angles.
- Say which angles are equal. In an isosceles triangle the two sides marked with ticks are equal. The angles opposite them (the base angles) are also equal.The two base angles are both x.
Reason: Base angles of an isosceles triangle are equal. - Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.48° + x + x = 180°
- Subtract the apex angle. This leaves the total of the two base angles.2x = 180 − 48 = 132°
- Divide by 2. The two base angles are equal, so each is half of the total.x = 132 ÷ 2 = 66°
Check. Add all three angles: 48 + 66 + 66 = 180°, which is 180°. ✓
Watch out. Share the remaining angle equally: divide by 2. Do not give each base angle the whole remainder.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Add them, then take the total away from 180°.
- Say which angles are equal. The ticks show two equal sides, so the base angles are equal. We are given one base angle, so the other one is the same.Other base angle = 38°
Reason: Base angles of an isosceles triangle are equal. - Add the two base angles. We need their total to take away from 180°.38 + 38 = 76°
- Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 76 = 104°
x = 104°
Check. Add all three angles: 38 + 38 + 104 = 180°, which is 180°. ✓
Watch out. Do not forget the second base angle: there are two equal angles, so subtract twice the base angle.
Key words. An exterior angle is formed outside a shape by extending one of its sides. The opposite interior angles are the two corners of the triangle that do not touch it.
Method. The exterior angle of a triangle equals the sum of the two opposite interior angles.
- Spot the exterior angle. An exterior angle is made when one side of the triangle is extended. The angle x is outside the triangle.x is an exterior angle.
- Use the exterior angle fact. The exterior angle equals the sum of the two interior angles that are not next to it (the opposite interior angles).The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- Add the two opposite angles. The two angles not next to x are 47° and 68°.x = 47 + 68 = 115°
Check. Another way: the third angle of the triangle is 180 − 47 − 68 = 65°, and on a straight line 65 + x = 180, so x = 115°. ✓
Watch out. Use the two angles opposite the exterior angle, not the angle next to it.
Watch out
The base angles of an isosceles triangle are the angles opposite the equal sides, which are at the ends of the third side (the base). Students often assume that the two angles next to the 50° are equal even when the ticks are on different sides. Always read the ticks.Exercise 17C Angles in triangles
Non-calculatorKey words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the triangle must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.50 + 65 = 115°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 115 = 65°
x = 65°
Check. Add all the angles: 50 + 65 + 65 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the triangle must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.38 + 72 = 110°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 110 = 70°
x = 70°
Check. Add all the angles: 38 + 72 + 70 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the triangle must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.24 + 105 = 129°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 129 = 51°
x = 51°
Check. Add all the angles: 24 + 105 + 51 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the triangle must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.34 + 90 = 124°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 124 = 56°
x = 56°
Check. Add all the angles: 34 + 90 + 56 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Take the apex angle away from 180° and share what is left between the two base angles.
- Say which angles are equal. In an isosceles triangle the two sides marked with ticks are equal. The angles opposite them (the base angles) are also equal.The two base angles are both x.
Reason: Base angles of an isosceles triangle are equal. - Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.40° + x + x = 180°
- Subtract the apex angle. This leaves the total of the two base angles.2x = 180 − 40 = 140°
- Divide by 2. The two base angles are equal, so each is half of the total.x = 140 ÷ 2 = 70°
Check. Add all three angles: 40 + 70 + 70 = 180°, which is 180°. ✓
Watch out. Share the remaining angle equally: divide by 2. Do not give each base angle the whole remainder.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Add them, then take the total away from 180°.
- Say which angles are equal. The ticks show two equal sides, so the base angles are equal. We are given one base angle, so the other one is the same.Other base angle = 65°
Reason: Base angles of an isosceles triangle are equal. - Add the two base angles. We need their total to take away from 180°.65 + 65 = 130°
- Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 130 = 50°
x = 50°
Check. Add all three angles: 65 + 65 + 50 = 180°, which is 180°. ✓
Watch out. Do not forget the second base angle: there are two equal angles, so subtract twice the base angle.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Take the apex angle away from 180° and share what is left between the two base angles.
- Say which angles are equal. In an isosceles triangle the two sides marked with ticks are equal. The angles opposite them (the base angles) are also equal.The two base angles are both x.
Reason: Base angles of an isosceles triangle are equal. - Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.100° + x + x = 180°
- Subtract the apex angle. This leaves the total of the two base angles.2x = 180 − 100 = 80°
- Divide by 2. The two base angles are equal, so each is half of the total.x = 80 ÷ 2 = 40°
Check. Add all three angles: 100 + 40 + 40 = 180°, which is 180°. ✓
Watch out. Share the remaining angle equally: divide by 2. Do not give each base angle the whole remainder.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Add them, then take the total away from 180°.
- Say which angles are equal. The ticks show two equal sides, so the base angles are equal. We are given one base angle, so the other one is the same.Other base angle = 35°
Reason: Base angles of an isosceles triangle are equal. - Add the two base angles. We need their total to take away from 180°.35 + 35 = 70°
- Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 70 = 110°
x = 110°
Check. Add all three angles: 35 + 35 + 110 = 180°, which is 180°. ✓
Watch out. Do not forget the second base angle: there are two equal angles, so subtract twice the base angle.
Method. An equilateral triangle has three equal sides, so it has three equal angles.
- State what we know. The ticks show that all three sides are equal, so all three angles are equal.Three equal angles
- Use the angle sum. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 ÷ 3 = 60°
- Write the answer. Equal angles each take a third of 180°.x = 60°
Check. 60 + 60 + 60 = 180°. ✓
Watch out. Every equilateral triangle has three 60° angles, whatever its size.
Key words. An exterior angle is formed outside a shape by extending one of its sides. The opposite interior angles are the two corners of the triangle that do not touch it.
Method. The exterior angle of a triangle equals the sum of the two opposite interior angles.
- Spot the exterior angle. An exterior angle is made when one side of the triangle is extended. The angle x is outside the triangle.x is an exterior angle.
- Use the exterior angle fact. The exterior angle equals the sum of the two interior angles that are not next to it (the opposite interior angles).The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- Add the two opposite angles. The two angles not next to x are 50° and 60°.x = 50 + 60 = 110°
Check. Another way: the third angle of the triangle is 180 − 50 − 60 = 70°, and on a straight line 70 + x = 180, so x = 110°. ✓
Watch out. Use the two angles opposite the exterior angle, not the angle next to it.
Key words. An exterior angle is formed outside a shape by extending one of its sides. The opposite interior angles are the two corners of the triangle that do not touch it.
Method. The exterior angle of a triangle equals the sum of the two opposite interior angles.
- Spot the exterior angle. An exterior angle is made when one side of the triangle is extended. The angle x is outside the triangle.x is an exterior angle.
- Use the exterior angle fact. The exterior angle equals the sum of the two interior angles that are not next to it (the opposite interior angles).The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- Add the two opposite angles. The two angles not next to x are 35° and 78°.x = 35 + 78 = 113°
Check. Another way: the third angle of the triangle is 180 − 35 − 78 = 67°, and on a straight line 67 + x = 180, so x = 113°. ✓
Watch out. Use the two angles opposite the exterior angle, not the angle next to it.
Method. Find the third angle, then compare each angle with 90°.
- Find angle C. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 70 − 45 = 65°
- Compare with 90°. A right-angled triangle has a 90° angle; an obtuse-angled one has an angle bigger than 90°.70°, 45° and 65° are all less than 90°
- Name the triangle. All three angles are acute.acute-angled
Check. 70 + 45 + 65 = 180°. ✓
Watch out. You must find the third angle before you can decide.
Key words. An isosceles triangle has two equal sides. The base angles are the two angles opposite the equal sides. The apex is the angle between the equal sides.
Method. The two base angles are equal. Take the apex angle away from 180° and share what is left between the two base angles.
- Say which angles are equal. In an isosceles triangle the two sides marked with ticks are equal. The angles opposite them (the base angles) are also equal.The two base angles are both x.
Reason: Base angles of an isosceles triangle are equal. - Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.110° + x + x = 180°
- Subtract the apex angle. This leaves the total of the two base angles.2x = 180 − 110 = 70°
- Divide by 2. The two base angles are equal, so each is half of the total.x = 70 ÷ 2 = 35°
Check. Add all three angles: 110 + 35 + 35 = 180°, which is 180°. ✓
Watch out. Share the remaining angle equally: divide by 2. Do not give each base angle the whole remainder.
Method. Share 180° in the ratio 1 : 2 : 3.
- Add the parts of the ratio. The ratio tells us how many equal parts the 180° is split into.1 + 2 + 3 = 6 parts
- Find one part. Divide 180° by 6.180 ÷ 6 = 30°
- Multiply to find each angle. 1 × 30 = 30°, 2 × 30 = 60°, 3 × 30 = 90°
- Name the triangle. One angle is 90°.right-angled
Check. 30 + 60 + 90 = 180°. ✓
Watch out. See Chapter 15 for sharing in a ratio.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 180°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Write an equation. The angles must add up to 180°, so we add all the expressions and put the answer equal to 180.x° + 2x° + 3x° = 180°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.6x = 180
- Divide both sides by 6. 6x means 6 × x, so dividing by 6 leaves x on its own.x = 180 ÷ 6 = 30
- Substitute x = 30 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.x° = 1 × 30 = 30°
2x° = 2 × 30 = 60°
3x° = 3 × 30 = 90°
Check. Add the angles: 30 + 60 + 90 = 180°, which is 180°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 180°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Write an equation. The angles must add up to 180°, so we add all the expressions and put the answer equal to 180.(2x + 10°) + 3x° + (x + 20°) = 180°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.6x + 30 = 180
- Get the number on the other side. Subtract 30 from both sides.6x = 150
- Divide both sides by 6. 6x means 6 × x, so dividing by 6 leaves x on its own.x = 150 ÷ 6 = 25
- Substitute x = 25 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.2x + 10° = 2 × 25 + 10 = 60°
3x° = 3 × 25 = 75°
x + 20° = 1 × 25 + 20 = 45°
Check. Add the angles: 60 + 75 + 45 = 180°, which is 180°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Method. Use the fact that the angles of a triangle add up to 180°.
- Suppose a triangle has two right angles. Each right angle is 90°.90° + 90° = 180°
- Look at what is left for the third angle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180° − 180° = 0°
- Decide. A corner of 0° is not a corner at all, so the shape would not be a triangle.It is impossible.
Check. A triangle must have three angles bigger than 0°. ✓
Watch out. This is a good example of a short argument that uses the angle sum fact.
Method. Add the angles and compare with 180°.
- Recall the fact. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.Angles in a triangle add up to 180°.
- Add. The total of the three angles should be 180°.70 + 60 + 60 = 190°
- Decide. 190 is bigger than 180.No, Sam cannot be correct.
Check. If the angles were 70°, 60° and 50° they would total 180°. ✓
Method. Find the interior angle at C using the straight line, then use the isosceles triangle.
- Find angle ACB. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 130 = 50°
- Say which angles are equal. AC = BC, so the angles opposite them, at B and at A, are equal.angle CAB = angle CBA
Reason: Base angles of an isosceles triangle are equal. - Use the angle sum. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 50 = 130° for the two base angles
- Divide by 2. The two base angles are equal.130 ÷ 2 = 65°
Check. 65 + 65 + 50 = 180°. ✓
Watch out. Another way: the exterior angle equals the sum of the two opposite angles: 65 + 65 = 130°.
Parallel lines
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Parallel lines are lines that stay the same distance apart for ever. We mark them with matching arrows. A line that crosses a pair of parallel lines is called a transversal. It makes eight angles, and they come in sets of equal angles or angles that add up to 180°. You do not need to learn eight facts: learn the three shapes below.
Alternate angles are equal (a Z shape). They lie between the parallel lines, on opposite sides of the transversal. Trace a Z (or an S) and you will find the two equal angles in its corners.
Corresponding angles are equal (an F shape). They are in the same position at the two crossings, like a copy of the same angle slid along the transversal. Trace an F (it may be backwards or upside down).
Co-interior angles add up to 180° (a C or U shape). They lie between the parallel lines on the same side of the transversal. They are not equal (unless both are 90°); instead they make a straight-line total of 180°. They are also called allied angles, or “interior angles on the same side”.
Why do these facts work? Corresponding angles are equal because sliding the top crossing down the transversal along the parallel lines moves one angle exactly onto the other. Alternate angles are equal because one of them is vertically opposite to the angle corresponding to the other. And co-interior angles add to 180° because one of them is next to an angle that is alternate to the other, on a straight line. These facts are only true when the lines are parallel. If the lines are not parallel, the angles are not equal.
How to answer. (1) Check for arrows. (2) Find a pair of angles making a Z, F or C shape (or an X or a straight line). (3) Write the reason in words. Often you need two steps: write the reason for each.
Key ideas
- Alternate angles are equal (Z shape).
- Corresponding angles are equal (F shape).
- Co-interior angles add up to 180° (C or U shape).
- These only work for parallel lines (look for the arrows).
- You can use vertically opposite angles and angles on a line as well, to reach the angle you need.
- Always write the reason in words: “alternate angles are equal”.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 65° angle = 65°
Wanted: x - Find which angle fact links the 65° angle and x. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the 65° angle = 65°
x = 65°
Check. Does the answer look right on the diagram? The angle marked x looks acute (less than 90°), and 65° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 48° angle = 48°
Wanted: y - Find which angle fact links the 48° angle and y. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They make a Z shape, so the fact is: Alternate angles are equal.
- Use the fact to find y. The two angles are equal, so the new angle is the same size as the known one.the 48° angle = 48°
y = 48°
Check. Does the answer look right on the diagram? The angle marked y looks acute (less than 90°), and 48° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 70° angle = 70°
Wanted: z - Find which angle fact links the 70° angle and z. The two angles are between the parallel lines on the same side of the transversal: a C or U shape (parallel lines).They make a C or U shape, so the fact is: Co-interior angles add up to 180°.
- Use the fact to find z. The two angles add up to 180°, so subtract the known angle from 180°.the 70° angle = 70°
180 − 70 = 110°
Check. Does the answer look right on the diagram? The angle marked z looks obtuse (between 90° and 180°), and 110° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 128° angle = 128°
Wanted: x
(this needs 2 steps) - Find which angle fact links the 128° angle and the angle at the top crossing, below the line and to the left of the transversal. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find the angle at the top crossing, below the line and to the left of the transversal. The two angles add up to 180°, so subtract the known angle from 180°.the 128° angle = 128°
180 − 128 = 52° - Find which angle fact links the angle at the top crossing, below the line and to the left of the transversal and x. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, below the line and to the left of the transversal = 52°
x = 52°
Check. Does the answer look right on the diagram? The angle marked x looks acute (less than 90°), and 52° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Watch out
Never use these facts unless the lines are parallel. Also, do not mix up the shapes: Z (alternate) and F (corresponding) angles are equal, but a C shape (co-interior) adds to 180°. A quick check: if both angles look the same size on the diagram, they are equal; if one is acute and the other obtuse, they add to 180°.Exercise 17D Parallel lines: alternate, corresponding and co-interior angles
Non-calculatorKey words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 65° angle = 65°
Wanted: x - Find which angle fact links the 65° angle and x. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the 65° angle = 65°
x = 65°
Check. Does the answer look right on the diagram? The angle marked x looks acute (less than 90°), and 65° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 72° angle = 72°
Wanted: y - Find which angle fact links the 72° angle and y. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They make a Z shape, so the fact is: Alternate angles are equal.
- Use the fact to find y. The two angles are equal, so the new angle is the same size as the known one.the 72° angle = 72°
y = 72°
Check. Does the answer look right on the diagram? The angle marked y looks acute (less than 90°), and 72° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 65° angle = 65°
Wanted: z - Find which angle fact links the 65° angle and z. The two angles are between the parallel lines on the same side of the transversal: a C or U shape (parallel lines).They make a C or U shape, so the fact is: Co-interior angles add up to 180°.
- Use the fact to find z. The two angles add up to 180°, so subtract the known angle from 180°.the 65° angle = 65°
180 − 65 = 115°
Check. Does the answer look right on the diagram? The angle marked z looks obtuse (between 90° and 180°), and 115° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 48° angle = 48°
Wanted: a - Find which angle fact links the 48° angle and a. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find a. The two angles are equal, so the new angle is the same size as the known one.the 48° angle = 48°
a = 48°
Check. Does the answer look right on the diagram? The angle marked a looks acute (less than 90°), and 48° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 133° angle = 133°
Wanted: b - Find which angle fact links the 133° angle and b. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They make a Z shape, so the fact is: Alternate angles are equal.
- Use the fact to find b. The two angles are equal, so the new angle is the same size as the known one.the 133° angle = 133°
b = 133°
Check. Does the answer look right on the diagram? The angle marked b looks obtuse (between 90° and 180°), and 133° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 56° angle = 56°
Wanted: c - Find which angle fact links the 56° angle and c. The two angles are between the parallel lines on the same side of the transversal: a C or U shape (parallel lines).They make a C or U shape, so the fact is: Co-interior angles add up to 180°.
- Use the fact to find c. The two angles add up to 180°, so subtract the known angle from 180°.the 56° angle = 56°
180 − 56 = 124°
Check. Does the answer look right on the diagram? The angle marked c looks obtuse (between 90° and 180°), and 124° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 139° angle = 139°
Wanted: x - Find which angle fact links the 139° angle and x. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They make a Z shape, so the fact is: Alternate angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the 139° angle = 139°
x = 139°
Check. Does the answer look right on the diagram? The angle marked x looks obtuse (between 90° and 180°), and 139° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 103° angle = 103°
Wanted: p - Find which angle fact links the 103° angle and p. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find p. The two angles are equal, so the new angle is the same size as the known one.the 103° angle = 103°
p = 103°
Check. Does the answer look right on the diagram? The angle marked p looks obtuse (between 90° and 180°), and 103° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 104° angle = 104°
Wanted: q - Find which angle fact links the 104° angle and q. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find q. The two angles are equal, so the new angle is the same size as the known one.the 104° angle = 104°
q = 104°
Check. Does the answer look right on the diagram? The angle marked q looks obtuse (between 90° and 180°), and 104° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 123° angle = 123°
Wanted: m - Find which angle fact links the 123° angle and m. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They make a Z shape, so the fact is: Alternate angles are equal.
- Use the fact to find m. The two angles are equal, so the new angle is the same size as the known one.the 123° angle = 123°
m = 123°
Check. Does the answer look right on the diagram? The angle marked m looks obtuse (between 90° and 180°), and 123° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle a is, then find the angle that makes the right shape with it.
- Find angle a on the diagram. Say where it is: the angle at the top crossing, above the line and to the right of the transversal.angle a = the angle at the top crossing, above the line and to the right of the transversal
- Look for the corresponding angle. Corresponding angles are in the same position at the other crossing point.The angles a and e make an F shape.
- Write the answer. angle e
Check. Both angles are marked on the diagram, and a and e are corresponding. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle c is, then find the angle that makes the right shape with it.
- Find angle c on the diagram. Say where it is: the angle at the top crossing, below the line and to the left of the transversal.angle c = the angle at the top crossing, below the line and to the left of the transversal
- Look for the alternate angle. Alternate angles are between the parallel lines on the opposite side of the transversal.The angles c and e make a Z shape.
- Write the answer. angle e
Check. Both angles are marked on the diagram, and c and e are alternate. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle d is, then find the angle that makes the right shape with it.
- Find angle d on the diagram. Say where it is: the angle at the top crossing, below the line and to the right of the transversal.angle d = the angle at the top crossing, below the line and to the right of the transversal
- Look for the co-interior angle. Co-interior angles are between the parallel lines on the same side of the transversal.The angles d and e make a C or U shape.
- Write the answer. angle e
Check. Both angles are marked on the diagram, and d and e are co-interior. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle d is, then find the angle that makes the right shape with it.
- Find angle d on the diagram. Say where it is: the angle at the top crossing, below the line and to the right of the transversal.angle d = the angle at the top crossing, below the line and to the right of the transversal
- Look for the corresponding angle. Corresponding angles are in the same position at the other crossing point.The angles d and h make an F shape.
- Write the answer. angle h
Check. Both angles are marked on the diagram, and d and h are corresponding. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle d is, then find the angle that makes the right shape with it.
- Find angle d on the diagram. Say where it is: the angle at the top crossing, below the line and to the right of the transversal.angle d = the angle at the top crossing, below the line and to the right of the transversal
- Look for the alternate angle. Alternate angles are between the parallel lines on the opposite side of the transversal.The angles d and f make a Z shape.
- Write the answer. angle f
Check. Both angles are marked on the diagram, and d and f are alternate. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle c is, then find the angle that makes the right shape with it.
- Find angle c on the diagram. Say where it is: the angle at the top crossing, below the line and to the left of the transversal.angle c = the angle at the top crossing, below the line and to the left of the transversal
- Look for the co-interior angle. Co-interior angles are between the parallel lines on the same side of the transversal.The angles c and f make a C or U shape.
- Write the answer. angle f
Check. Both angles are marked on the diagram, and c and f are co-interior. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: b - Find which angle fact links the 62° angle and b. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find b. The two angles add up to 180°, so subtract the known angle from 180°.the 62° angle = 62°
180 − 62 = 118°
Check. Does the answer look right on the diagram? The angle marked b looks obtuse (between 90° and 180°), and 118° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: c - Find which angle fact links the 62° angle and c. The two angles are opposite each other where two straight lines cross (an X shape).They make an X shape, so the fact is: Vertically opposite angles are equal.
- Use the fact to find c. The two angles are equal, so the new angle is the same size as the known one.the 62° angle = 62°
c = 62°
Check. Does the answer look right on the diagram? The angle marked c looks acute (less than 90°), and 62° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: d - Find which angle fact links the 62° angle and d. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find d. The two angles add up to 180°, so subtract the known angle from 180°.the 62° angle = 62°
180 − 62 = 118°
Check. Does the answer look right on the diagram? The angle marked d looks obtuse (between 90° and 180°), and 118° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: e - Find which angle fact links the 62° angle and e. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find e. The two angles are equal, so the new angle is the same size as the known one.the 62° angle = 62°
e = 62°
Check. Does the answer look right on the diagram? The angle marked e looks acute (less than 90°), and 62° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: f
(this needs 2 steps) - Find which angle fact links the 62° angle and the angle at the top crossing, above the line and to the left of the transversal. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find the angle at the top crossing, above the line and to the left of the transversal. The two angles add up to 180°, so subtract the known angle from 180°.the 62° angle = 62°
180 − 62 = 118° - Find which angle fact links the angle at the top crossing, above the line and to the left of the transversal and f. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find f. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, above the line and to the left of the transversal = 118°
f = 118°
Check. Does the answer look right on the diagram? The angle marked f looks obtuse (between 90° and 180°), and 118° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: g
(this needs 2 steps) - Find which angle fact links the 62° angle and the angle at the top crossing, below the line and to the left of the transversal. The two angles are opposite each other where two straight lines cross (an X shape).They make an X shape, so the fact is: Vertically opposite angles are equal.
- Use the fact to find the angle at the top crossing, below the line and to the left of the transversal. The two angles are equal, so the new angle is the same size as the known one.the 62° angle = 62°
the angle at the top crossing, below the line and to the left of the transversal = 62° - Find which angle fact links the angle at the top crossing, below the line and to the left of the transversal and g. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find g. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, below the line and to the left of the transversal = 62°
g = 62°
Check. Does the answer look right on the diagram? The angle marked g looks acute (less than 90°), and 62° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: h
(this needs 2 steps) - Find which angle fact links the 62° angle and the angle at the top crossing, below the line and to the right of the transversal. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find the angle at the top crossing, below the line and to the right of the transversal. The two angles add up to 180°, so subtract the known angle from 180°.the 62° angle = 62°
180 − 62 = 118° - Find which angle fact links the angle at the top crossing, below the line and to the right of the transversal and h. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find h. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, below the line and to the right of the transversal = 118°
h = 118°
Check. Does the answer look right on the diagram? The angle marked h looks obtuse (between 90° and 180°), and 118° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Method. Recall the three facts for parallel lines.
- Recall the fact. Alternate and corresponding angles are equal; co-interior angles add up to 180°.Co-interior angles add up to 180°.
- Compare. Co-interior angles are equal only when both are 90°.The statement is false in general.
Check. For 70° and 110°: they add to 180 but are not equal. ✓
Method. Recall when the facts work.
- Recall the condition. Alternate angles are equal only if the two lines are parallel.Alternate angles are equal. (parallel lines)
- Decide. If the lines are not parallel, the angles are different.False
Check. Draw two lines that meet far away: the alternate angles are not equal. ✓
Method. Identify the correct pair of angles by their shape.
- Look at the shape made by the two angles. The 70° angle and x are between the parallel lines on opposite sides of the transversal: a Z shape.They are alternate angles.
- Spot Ali’s mistake. Ali used the co-interior rule (180°) but these are not co-interior angles.wrong fact
- Use the correct fact. Alternate angles are equal.x = 70°
Check. On the diagram both angles look acute, so they are equal, not 110°. ✓
Watch out. Choose the fact by the shape (Z, F or C).
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Decide which angle fact connects the two angles, form an equation, solve it, then substitute to find the angles.
- Say how the two angles are related. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They are alternate angles, so they are equal. Reason: Alternate angles are equal.
- Write the equation. The two expressions are equal (or the angles add to the right total), so we can write an equation and solve it.2x + 30 = x + 40
- Collect the x terms on one side. Subtract x from both sides, so that there is only one x term left.x + 30 = 40
- Collect the numbers on the other side. Subtract 30 from both sides.x = 10
- Read off the value of x. x = 10
- Substitute x = 10 to find the first angle. Now we know the value of x, put it into the expression for the angle.2x + 30 = 2 × 10 + 30 = 50°
- Substitute x = 10 to find the second angle. Now we know the value of x, put it into the expression for the angle.x + 40 = 1 × 10 + 40 = 50°
Check. Check: the two angles are 50° and 50°; they are equal. ✓
Watch out. Substitute back into the expressions to check. Then make sure you answer what was asked: the value of x, or the size of an angle.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Decide which angle fact connects the two angles, form an equation, solve it, then substitute to find the angles.
- Say how the two angles are related. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They are corresponding angles, so they are equal. Reason: Corresponding angles are equal.
- Write the equation. The two expressions are equal (or the angles add to the right total), so we can write an equation and solve it.2x + 60 = 3x + 40
- Collect the x terms on one side. Subtract 2x from both sides (the smaller x term), which keeps the number in front of x positive.60 = x + 40
- Collect the numbers on the other side. Subtract 40 from both sides.20 = x
- Read off the value of x. x = 20
- Substitute x = 20 to find the first angle. Now we know the value of x, put it into the expression for the angle.2x + 60 = 2 × 20 + 60 = 100°
- Substitute x = 20 to find the second angle. Now we know the value of x, put it into the expression for the angle.3x + 40 = 3 × 20 + 40 = 100°
Check. Check: the two angles are 100° and 100°; they are equal. ✓
Watch out. Substitute back into the expressions to check. Then make sure you answer what was asked: the value of x, or the size of an angle.
Method. Use the angle facts one at a time, starting from the 90° angle.
- Use angles on a straight line. Angles on a straight line add up to 180°.The angle next to 90° is 180 − 90 = 90°
- Use vertically opposite angles. Vertically opposite angles are equal.The angle opposite a 90° angle is 90°
- Use corresponding angles. Corresponding angles are equal.The angles at the other crossing are copies, so they are 90° as well
- State the conclusion. Every angle is reached by one of these facts.All eight angles are 90°: the transversal is perpendicular to both lines.
Check. If the transversal is perpendicular to one of the lines, it is perpendicular to the other too. ✓
Exercise 17E Parallel lines: multi-step angle problems and reasons
Non-calculatorKey words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 128° angle = 128°
Wanted: x
(this needs 2 steps) - Find which angle fact links the 128° angle and the angle at the top crossing, below the line and to the left of the transversal. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find the angle at the top crossing, below the line and to the left of the transversal. The two angles add up to 180°, so subtract the known angle from 180°.the 128° angle = 128°
180 − 128 = 52° - Find which angle fact links the angle at the top crossing, below the line and to the left of the transversal and x. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, below the line and to the left of the transversal = 52°
x = 52°
Check. Does the answer look right on the diagram? The angle marked x looks acute (less than 90°), and 52° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: y
(this needs 2 steps) - Find which angle fact links the 62° angle and the angle at the bottom crossing, above the line and to the left of the transversal. The two angles are opposite each other where two straight lines cross (an X shape).They make an X shape, so the fact is: Vertically opposite angles are equal.
- Use the fact to find the angle at the bottom crossing, above the line and to the left of the transversal. The two angles are equal, so the new angle is the same size as the known one.the 62° angle = 62°
the angle at the bottom crossing, above the line and to the left of the transversal = 62° - Find which angle fact links the angle at the bottom crossing, above the line and to the left of the transversal and y. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find y. The two angles are equal, so the new angle is the same size as the known one.the angle at the bottom crossing, above the line and to the left of the transversal = 62°
y = 62°
Check. Does the answer look right on the diagram? The angle marked y looks acute (less than 90°), and 62° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. PAQ is parallel to BC, so angles PAB and ABC are alternate angles (a Z shape).
- Spot the pair. The angle at B and the angle at A are between the parallel lines on opposite sides of AB.angle PAB and angle ABC form a Z shape
- Use the fact. Alternate angles are equal.angle PAB = angle ABC = 55°
Check. Both angles look acute and the same size on the diagram. ✓
Method. The angles PAB, BAC and QAC all lie on the straight line PAQ, so they add up to 180°.
- Write the sum. Angles on a straight line add up to 180°.55 + angle BAC + 65 = 180
- Add and subtract. 55 + 65 = 120.180 − 120 = 60°
Check. The angles of triangle ABC: 55 + 60 + 65 = 180°. ✓
Method. Draw an extra line through P that is parallel to the other two lines, then use alternate angles twice.
- Draw the extra line. This line through P is parallel to the other two, so we can use the parallel-line facts for it.a third parallel line through P
- Find the upper part. The upper part of angle x is alternate to the 35° angle at A.upper part = 35° (alternate angles are equal)
- Find the lower part. The lower part of angle x is alternate to the 48° angle at C.lower part = 48° (alternate angles are equal)
- Add the two parts. Together they make angle APC.35 + 48 = 83°
Check. Test the pattern: the angle at P should be bigger than each of the two outside angles, and 83 is bigger than 35 and bigger than 48. ✓
Watch out. Drawing a parallel line through the point in the middle is a very useful trick.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 130° angle = 130°
Wanted: x
(this needs 2 steps) - Find which angle fact links the 130° angle and the angle at the top crossing, below the line and to the right of the transversal. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find the angle at the top crossing, below the line and to the right of the transversal. The two angles add up to 180°, so subtract the known angle from 180°.the 130° angle = 130°
180 − 130 = 50° - Find which angle fact links the angle at the top crossing, below the line and to the right of the transversal and x. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, below the line and to the right of the transversal = 50°
x = 50°
Check. Does the answer look right on the diagram? The angle marked x looks acute (less than 90°), and 50° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Method. Add the two alternate angles at P (use a third parallel line through P).
- Use alternate angles for the top part. Alternate angles are equal.angle between PA and the parallel line = 40°
- Use alternate angles for the bottom part. Alternate angles are equal.angle between PC and the parallel line = 55°
- Add. Angle APC is made of the two parts.40 + 55 = 95°
Check. This agrees with the pattern: the middle angle equals the sum of the two outside angles. ✓
Quadrilaterals
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
A quadrilateral is a polygon with four straight sides. Draw any quadrilateral and cut it in two with a diagonal: you get two triangles. Each triangle has angles adding up to 180°, so the four angles of any quadrilateral add up to 2 × 180° = 360°.
Some quadrilaterals have special names and special properties. You must know them all, and use them to find angles and sides. In these diagrams, matching ticks show equal sides, arrows show parallel sides and a small square shows a right angle.
The table summarises the properties. The last two columns (lines of symmetry and order of rotational symmetry) are worked out from the shapes themselves; see Section 17.7.
| Quadrilateral | Sides | Angles | Diagonals | Lines of symmetry | Order of rotation |
|---|---|---|---|---|---|
| Square | 4 equal sides; opposite sides parallel | all 90° | equal, bisect each other at 90° | 4 | 4 |
| Rectangle | opposite sides equal and parallel | all 90° | equal, bisect each other | 2 | 2 |
| Parallelogram | opposite sides equal and parallel | opposite angles equal | bisect each other | 0 | 2 |
| Rhombus | 4 equal sides; opposite sides parallel | opposite angles equal | bisect each other at 90°; bisect the angles | 2 | 2 |
| Kite | two pairs of equal adjacent sides | one pair of equal angles (opposite each other) | cross at 90° | 1 | 1 |
| Isosceles trapezium | one pair of parallel sides; the other two sides equal | two pairs of equal angles (base angles) | equal | 1 | 1 |
| Trapezium | exactly one pair of parallel sides | angles between the parallel sides add to 180° | no special property | 0 | 1 |
Notice the family tree: a square is a special rectangle (all sides equal), a special rhombus (all angles 90°) and a special parallelogram. A rectangle and a rhombus are both special parallelograms. A parallelogram has two pairs of parallel sides; a trapezium has just one pair.
Using the properties for angles. In a parallelogram, opposite angles are equal and neighbouring angles add up to 180° (they are co-interior angles between the parallel sides). In a kite, one pair of opposite angles (the ones between the unequal sides) are equal. In a trapezium, the two angles at the ends of a non-parallel side add up to 180°. In an isosceles trapezium, the base angles are equal.
Key ideas
- Angles in a quadrilateral add up to 360°.
- Parallelogram: opposite sides parallel and equal; opposite angles equal; neighbouring angles add to 180°.
- Rhombus: a parallelogram with 4 equal sides; diagonals cross at 90° and bisect the angles.
- Rectangle: four right angles; diagonals equal. Square: all sides equal and all angles 90°.
- Kite: two pairs of equal adjacent sides; diagonals cross at 90°.
- Trapezium: one pair of parallel sides (isosceles: the other two sides equal).
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the quadrilateral must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.85 + 100 + 75 = 260°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 260 = 100°
x = 100°
Check. Add all the angles: 85 + 100 + 75 + 100 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use the properties of a parallelogram: opposite angles are equal and neighbouring angles add to 180°.
- Find x. Angles A and B are neighbours, between the parallel sides AD and BC (co-interior angles).x = 180 − 68 = 112° (co-interior angles add up to 180°)
- Find y. Angle C is opposite angle A.y = 68° (opposite angles of a parallelogram are equal)
- Find z. Angle D is opposite angle B.z = 112° (opposite angles of a parallelogram are equal)
Check. Add all four angles: 68 + 112 + 68 + 112 = 360°. ✓
Watch out. Opposite angles are equal; neighbouring angles add to 180°.
Key words. A kite has two pairs of equal sides next to each other. It has one line of symmetry.
Method. A kite has one pair of equal angles, at B and D. Use the angle sum 360°, then share what is left.
- Say which angles are equal. In a kite the angles between the unequal sides are equal. Here angle ABC = angle ADC, because the two triangles on either side of AC match.angle ABC = angle ADC = x (a kite has one pair of equal angles)
- Use the angle sum of a quadrilateral. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.100 + 60 + x + x = 360
- Collect and subtract. 100 + 60 = 160, and the two equal angles are what is left.2x = 360 − 160 = 200
- Divide by 2. x = 200 ÷ 2 = 100°
Check. Add all four angles: 100 + 100 + 60 + 100 = 360°. ✓
Watch out. The equal angles in a kite are the ones between a short side and a long side, not the ones at the ends of the line of symmetry.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. The angles at the ends of a non-parallel side of a trapezium are co-interior angles, so they add up to 180°.
- Spot the pair of angles. AB and DC are parallel, and AD joins them. So angle DAB and angle ADC are co-interior angles.angle DAB and angle ADC lie between parallel lines on the same side of AD
- State the fact. Co-interior angles add up to 180°.65 + x = 180
- Subtract. x = 180 − 65 = 115°
Check. The two base angles at A and B are both 65° and the two at C and D are both 115°: 65 + 65 + 115 + 115 = 360°. ✓
Watch out. In an isosceles trapezium the angles on each parallel side are equal.
Watch out
A square is also a rectangle, a rhombus and a parallelogram, and a rectangle is also a parallelogram. When you are asked to “name a quadrilateral” give the most specific name that fits. Also, do not mix up a parallelogram (opposite angles equal) with a kite (only one pair of equal angles).Exercise 17F Quadrilaterals
Non-calculatorKey words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the quadrilateral must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.85 + 100 + 75 = 260°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 260 = 100°
x = 100°
Check. Add all the angles: 85 + 100 + 75 + 100 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the quadrilateral must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.110 + 90 + 70 = 270°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 270 = 90°
x = 90°
Check. Add all the angles: 110 + 90 + 70 + 90 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the quadrilateral must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.90 + 90 + 125 = 305°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 305 = 55°
x = 55°
Check. Add all the angles: 90 + 90 + 125 + 55 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the quadrilateral must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.65 + 130 + 82 = 277°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 277 = 83°
y = 83°
Check. Add all the angles: 65 + 130 + 82 + 83 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.square: a quadrilateral with four equal sides and four right angles
- Match the description to the word. All sides are equal and all angles are 90°.The answer is square.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.parallelogram: a quadrilateral with two pairs of parallel sides (opposite sides equal and parallel)
- Match the description to the word. Opposite sides are parallel and equal, and the angles are not 90°.The answer is parallelogram.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.rhombus: a quadrilateral with four equal sides (a squashed square)
- Match the description to the word. All sides are equal but the angles are not 90°.The answer is rhombus.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.trapezium: a quadrilateral with exactly one pair of parallel sides
- Match the description to the word. Only one pair of sides is parallel.The answer is trapezium.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.kite: a quadrilateral with two pairs of equal adjacent sides; one line of symmetry
- Match the description to the word. The equal sides are next to each other, not opposite.The answer is kite.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.rectangle: a quadrilateral with four right angles (opposite sides equal)
- Match the description to the word. All angles are 90° but the sides are not all equal.The answer is rectangle.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. In a parallelogram, angles that share a side are co-interior angles between parallel sides, so they add up to 180°.
- Spot the relationship. Angles A and B are at the two ends of side AB. The sides AD and BC are parallel, so A and B are co-interior angles.A and B are co-interior angles
- State the fact. Co-interior angles add up to 180°.A + B = 180°
- Subtract. x = 180 − 68 = 112°
Check. The angles of the parallelogram: 68 + 112 + 68 + 112 = 360°. ✓
Watch out. Neighbouring angles of a parallelogram add to 180°; opposite angles are equal.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. The opposite angles of a parallelogram are equal.
- Spot the relationship. Angle C is opposite angle A, across the parallelogram.C is opposite A
- State the fact. Opposite angles of a parallelogram are equal.Reason: opposite angles of a parallelogram are equal
- Write the answer. y = 68°
Check. Using co-interior angles instead: y + 112 = 180, so y = 68°. ✓
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. The opposite angles of a parallelogram are equal.
- Spot the relationship. Angle D is opposite angle B.D is opposite B, and B = 112°
- State the fact. Opposite angles of a parallelogram are equal.Reason: opposite angles of a parallelogram are equal
- Write the answer. z = 112°
Check. Angles A and D are co-interior: 68 + 112 = 180°. ✓
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. In a parallelogram, angles that share a side are co-interior angles between parallel sides, so they add up to 180°.
- Spot the relationship. Angles A and B are at the two ends of side AB. The sides AD and BC are parallel, so A and B are co-interior angles.A and B are co-interior angles
- State the fact. Co-interior angles add up to 180°.A + B = 180°
- Subtract. x = 180 − 125 = 55°
Check. The angles of the parallelogram: 125 + 55 + 125 + 55 = 360°. ✓
Watch out. Neighbouring angles of a parallelogram add to 180°; opposite angles are equal.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. The opposite angles of a parallelogram are equal.
- Spot the relationship. Angle C is opposite angle A, across the parallelogram.C is opposite A
- State the fact. Opposite angles of a parallelogram are equal.Reason: opposite angles of a parallelogram are equal
- Write the answer. y = 125°
Check. Using co-interior angles instead: y + 55 = 180, so y = 125°. ✓
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. The opposite angles of a parallelogram are equal.
- Spot the relationship. Angle D is opposite angle B.D is opposite B, and B = 55°
- State the fact. Opposite angles of a parallelogram are equal.Reason: opposite angles of a parallelogram are equal
- Write the answer. z = 55°
Check. Angles A and D are co-interior: 125 + 55 = 180°. ✓
Method. The two angles at B and D are equal. Take the known angles from 360° and share what is left.
- Say which angles are equal. In a kite the two angles between the unequal sides are equal.angle ABC = angle ADC
Reason: a kite has one pair of equal angles - Use the angle sum. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.110 + 50 + x + x = 360
- Subtract. 110 + 50 = 160.2x = 360 − 160 = 200
- Divide by 2. x = 100°
Check. 110 + 100 + 50 + 100 = 360°. ✓
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Angles B and C are at the ends of the side BC, which joins the parallel sides, so they are co-interior angles.
- Spot the pair. AB and DC are parallel and BC crosses them.angle B and angle C are co-interior
- State the fact. Co-interior angles add up to 180°.70 + x = 180
- Subtract. x = 180 − 70 = 110°
Check. The angles at C and B look obtuse and acute, adding to a straight line. ✓
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Angles A and D are at the ends of the side AD, which joins the parallel sides, so they are co-interior angles.
- Spot the pair. AB and DC are parallel and AD crosses them.angle A and angle D are co-interior
- State the fact. Co-interior angles add up to 180°.75 + y = 180
- Subtract. y = 180 − 75 = 105°
Check. 75 + 70 + 110 + 105 = 360°. ✓
Method. Triangle ABC is isosceles because the sides of a rhombus are equal. Use the angle sum.
- Spot the isosceles triangle. All four sides of a rhombus are equal, so AB = BC.AB = BC, so triangle ABC is isosceles
- Say which angles are equal. Base angles of an isosceles triangle are equal.angle BAC = angle BCA
- Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 110 = 70° for the two base angles
- Divide by 2. 70 ÷ 2 = 35°
Check. The diagonal of a rhombus bisects the angle: angle BAD = 180 − 110 = 70°, and 70 ÷ 2 = 35°. ✓
Watch out. The angle between a side and a diagonal of a rhombus is half the corner angle.
Method. Triangle ABC has a right angle at B, so use the angle sum of a triangle.
- Find the right angle. A rectangle has four right angles, so angle ABC = 90°.angle ABC = 90°
- Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.37 + 90 + x = 180
- Subtract. 37 + 90 = 127.x = 180 − 127 = 53°
Check. The two acute angles of a right-angled triangle add to 90°: 37 + 53 = 90. ✓
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.square
- Count the fold lines. Two diagonals, and two lines joining the middle of opposite sides.4
Check. Lines of symmetry of a square: 4. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.rectangle
- Count the fold lines. Two lines joining the middle of opposite sides. The diagonals do not work.2
Check. Lines of symmetry of a rectangle: 2. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.parallelogram
- Count the fold lines. Slanted sides stop any fold from matching. (It does have rotational symmetry.)0
Check. Lines of symmetry of a parallelogram: 0. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.rhombus
- Count the fold lines. The two diagonals. Lines joining the middle of opposite sides do not work.2
Check. Lines of symmetry of a rhombus: 2. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.kite
- Count the fold lines. Only the diagonal between the two pairs of equal sides.1
Check. Lines of symmetry of a kite: 1. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.isosceles trapezium
- Count the fold lines. One line through the middle of the two parallel sides.1
Check. Lines of symmetry of a isosceles trapezium: 1. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Compare the definitions.
- Recall a square. A square has four equal sides and four right angles.square: equal sides + right angles
- Recall a rectangle. A rectangle only needs four right angles.rectangle: right angles
- Decide. A 3 cm by 5 cm rectangle has right angles but unequal sides.False
Check. A square is a rectangle, but a rectangle is not always a square. ✓
Method. Compare the definitions.
- Recall a parallelogram. Two pairs of parallel sides.opposite sides parallel
- Check a rhombus. The opposite sides of a rhombus are parallel.it meets the definition
- Decide. True
Check. A rhombus is a parallelogram with equal sides. ✓
Method. Recall the definition.
- Recall a trapezium. It has exactly one pair of parallel sides.one pair
- Compare. Two pairs of parallel sides is a parallelogram.False
Check. A trapezium has only one pair of parallel sides. ✓
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 360°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Write an equation. The angles must add up to 360°, so we add all the expressions and put the answer equal to 360.x° + 2x° + 3x° + 4x° = 360°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.10x = 360
- Divide both sides by 10. 10x means 10 × x, so dividing by 10 leaves x on its own.x = 360 ÷ 10 = 36
- Substitute x = 36 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.x° = 1 × 36 = 36°
2x° = 2 × 36 = 72°
3x° = 3 × 36 = 108°
4x° = 4 × 36 = 144°
Check. Add the angles: 36 + 72 + 108 + 144 = 360°, which is 360°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 360°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Write an equation. The angles must add up to 360°, so we add all the expressions and put the answer equal to 360.(2x + 10°) + 3x° + (x + 20°) + (4x − 10°) = 360°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.10x + 20 = 360
- Get the number on the other side. Subtract 20 from both sides.10x = 340
- Divide both sides by 10. 10x means 10 × x, so dividing by 10 leaves x on its own.x = 340 ÷ 10 = 34
- Substitute x = 34 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.2x + 10° = 2 × 34 + 10 = 78°
3x° = 3 × 34 = 102°
x + 20° = 1 × 34 + 20 = 54°
4x − 10° = 4 × 34 − 10 = 126°
Check. Add the angles: 78 + 102 + 54 + 126 = 360°, which is 360°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Method. Use the angle sum 360° for the fourth angle, then think about which quadrilateral has four right angles.
- Add the three known angles. We need their total.90 + 90 + 90 = 270°
- Subtract from 360°. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.360 − 270 = 90°
- Name the quadrilateral. A quadrilateral with four right angles is a rectangle (or a square).rectangle
Check. 4 × 90 = 360°. ✓
Watch out. We cannot say it is a square: the sides might not be equal.
Interior and exterior angles of polygons
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
A polygon is a closed shape with straight sides. The angle inside the shape at a corner is the interior angle. If you extend one side beyond a corner, the angle between the extension and the next side is the exterior angle. At every corner, the interior angle and exterior angle sit on a straight line, so interior + exterior = 180°.
Sum of the interior angles. Choose one corner of a polygon and join it to every other corner you can. This cuts the polygon into triangles. A quadrilateral makes 2 triangles, a pentagon makes 3, a hexagon makes 4, and in general an n-sided polygon makes n − 2 triangles. Each triangle has an angle sum of 180°, so
sum of interior angles = (n − 2) × 180°
| Polygon | Sides n | Triangles n − 2 | Angle sum |
|---|---|---|---|
| Triangle | 3 | 1 | 180° |
| Quadrilateral | 4 | 2 | 360° |
| Pentagon | 5 | 3 | 540° |
| Hexagon | 6 | 4 | 720° |
| Heptagon | 7 | 5 | 900° |
| Octagon | 8 | 6 | 1080° |
| Nonagon | 9 | 7 | 1260° |
| Decagon | 10 | 8 | 1440° |
Regular polygons. In a regular polygon all the sides and all the angles are equal. So each interior angle is the sum divided by n: (n − 2) × 180° ÷ n. For a regular hexagon this is 720 ÷ 6 = 120°.
Exterior angles. Imagine walking round a polygon. At each corner you turn through the exterior angle. By the time you are back where you started and facing the same way, you have made exactly one full turn. So the exterior angles of any polygon add up to 360°. In a regular polygon they are all equal, so each exterior angle is 360° ÷ n. For a regular hexagon: 360 ÷ 6 = 60°, and the interior angle is 180 − 60 = 120°. This shortcut is often the quickest method.
Key ideas
- Interior angle + exterior angle = 180° at each corner.
- Sum of interior angles = (n − 2) × 180°.
- Sum of exterior angles = 360° (for every polygon).
- Regular polygon: each exterior angle = 360° ÷ n; each interior angle = 180° − exterior angle.
- To find the number of sides of a regular polygon: n = 360 ÷ exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 8 sides, so it is a octagon.n = 8
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.8 − 2 = 6 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.6 × 180 = 1080°
Check. Check with the formula (n − 2) × 180 = (8 − 2) × 180 = 1080°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular nonagon has n = 9 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(9 − 2) × 180 = 7 × 180 = 1260°
- Share the sum equally. In a regular polygon all 9 interior angles are equal, so each one is the sum divided by 9.1260 ÷ 9 = 140°
Check. Another way: each exterior angle is 360 ÷ 9 = 40°, and 180 − 40 = 140°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a pentagon add up to (5 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 5 sides, so it is a pentagon.n = 5
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.5 − 2 = 3 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.3 × 180 = 540°
- Add the angles we know. We need their total to take from the sum.100 + 120 + 90 + 110 = 420°
- Subtract from 540°. What is left is the unknown angle.540 − 420 = 120°
x = 120°
Check. Add all the angles: 100 + 120 + 90 + 110 + 120 = 540°, which is 540°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 24°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 24°, so the number of them is 360 ÷ 24.360 ÷ 24 = 15 sides
Check. 15 × 24 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Watch out
Do not divide 180° by the number of sides to get the interior angle, and do not mix up sum and each. The formula (n − 2) × 180° gives the sum of all the interior angles; you only divide by n for a regular polygon. Also, the exterior angles always add to 360°, whatever the polygon.Exercise 17G Angles in polygons: sums and missing angles
Non-calculatorKey words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 5 sides, so it is a pentagon.n = 5
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.5 − 2 = 3 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.3 × 180 = 540°
Check. Check with the formula (n − 2) × 180 = (5 − 2) × 180 = 540°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 6 sides, so it is a hexagon.n = 6
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.6 − 2 = 4 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.4 × 180 = 720°
Check. Check with the formula (n − 2) × 180 = (6 − 2) × 180 = 720°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 8 sides, so it is a octagon.n = 8
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.8 − 2 = 6 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.6 × 180 = 1080°
Check. Check with the formula (n − 2) × 180 = (8 − 2) × 180 = 1080°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 10 sides, so it is a decagon.n = 10
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.10 − 2 = 8 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.8 × 180 = 1440°
Check. Check with the formula (n − 2) × 180 = (10 − 2) × 180 = 1440°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 12 sides, so it is a dodecagon.n = 12
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.12 − 2 = 10 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.10 × 180 = 1800°
Check. Check with the formula (n − 2) × 180 = (12 − 2) × 180 = 1800°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a pentagon add up to (5 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 5 sides, so it is a pentagon.n = 5
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.5 − 2 = 3 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.3 × 180 = 540°
- Add the angles we know. We need their total to take from the sum.100 + 120 + 90 + 110 = 420°
- Subtract from 540°. What is left is the unknown angle.540 − 420 = 120°
x = 120°
Check. Add all the angles: 100 + 120 + 90 + 110 + 120 = 540°, which is 540°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a hexagon add up to (6 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 6 sides, so it is a hexagon.n = 6
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.6 − 2 = 4 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.4 × 180 = 720°
- Add the angles we know. We need their total to take from the sum.130 + 100 + 115 + 125 + 95 = 565°
- Subtract from 720°. What is left is the unknown angle.720 − 565 = 155°
y = 155°
Check. Add all the angles: 130 + 100 + 115 + 125 + 95 + 155 = 720°, which is 720°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a heptagon add up to (7 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 7 sides, so it is a heptagon.n = 7
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.7 − 2 = 5 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.5 × 180 = 900°
- Add the angles we know. We need their total to take from the sum.150 + 110 + 120 + 140 + 125 + 135 = 780°
- Subtract from 900°. What is left is the unknown angle.900 − 780 = 120°
z = 120°
Check. Add all the angles: 150 + 110 + 120 + 140 + 125 + 135 + 120 = 900°, which is 900°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a pentagon add up to (5 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 5 sides, so it is a pentagon.n = 5
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.5 − 2 = 3 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.3 × 180 = 540°
- Add the angles we know. We need their total to take from the sum.128 + 98 + 112 + 102 = 440°
- Subtract from 540°. What is left is the unknown angle.540 − 440 = 100°
a = 100°
Check. Add all the angles: 128 + 100 + 98 + 112 + 102 = 540°, which is 540°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a hexagon add up to (6 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 6 sides, so it is a hexagon.n = 6
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.6 − 2 = 4 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.4 × 180 = 720°
- Add the angles we know. We need their total to take from the sum.105 + 120 + 135 + 95 + 140 = 595°
- Subtract from 720°. What is left is the unknown angle.720 − 595 = 125°
b = 125°
Check. Add all the angles: 105 + 120 + 135 + 95 + 140 + 125 = 720°, which is 720°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Use the formula (n − 2) × 180 = sum, and work backwards.
- Write the formula as an equation. The interior angles of an n-sided polygon add up to (n − 2) × 180°.(n − 2) × 180 = 1260
- Divide both sides by 180. This undoes the multiplication by 180.n − 2 = 1260 ÷ 180 = 7
- Add 2 to both sides. This undoes the subtraction.n = 7 + 2 = 9
Check. (9 − 2) × 180 = 7 × 180 = 1260°. ✓
Watch out. Remember to add the 2 back after dividing.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular pentagon has n = 5 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(5 − 2) × 180 = 3 × 180 = 540°
- Share the sum equally. In a regular polygon all 5 interior angles are equal, so each one is the sum divided by 5.540 ÷ 5 = 108°
Check. Another way: each exterior angle is 360 ÷ 5 = 72°, and 180 − 72 = 108°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular hexagon has n = 6 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(6 − 2) × 180 = 4 × 180 = 720°
- Share the sum equally. In a regular polygon all 6 interior angles are equal, so each one is the sum divided by 6.720 ÷ 6 = 120°
Check. Another way: each exterior angle is 360 ÷ 6 = 60°, and 180 − 60 = 120°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular octagon has n = 8 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(8 − 2) × 180 = 6 × 180 = 1080°
- Share the sum equally. In a regular polygon all 8 interior angles are equal, so each one is the sum divided by 8.1080 ÷ 8 = 135°
Check. Another way: each exterior angle is 360 ÷ 8 = 45°, and 180 − 45 = 135°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular decagon has n = 10 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(10 − 2) × 180 = 8 × 180 = 1440°
- Share the sum equally. In a regular polygon all 10 interior angles are equal, so each one is the sum divided by 10.1440 ÷ 10 = 144°
Check. Another way: each exterior angle is 360 ÷ 10 = 36°, and 180 − 36 = 144°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular dodecagon has n = 12 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(12 − 2) × 180 = 10 × 180 = 1800°
- Share the sum equally. In a regular polygon all 12 interior angles are equal, so each one is the sum divided by 12.1800 ÷ 12 = 150°
Check. Another way: each exterior angle is 360 ÷ 12 = 30°, and 180 − 30 = 150°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Method. The sum must equal (n − 2) × 180 for a whole number n.
- Write the equation. The interior angles of an n-sided polygon add up to (n − 2) × 180°.(n − 2) × 180 = 900
- Divide by 180. n − 2 = 900 ÷ 180 = 5
- Decide. A whole number of sides (5 + 2 = 7) works, so the answer is yes.n = 7: a heptagon
Check. (7 − 2) × 180 = 900°. ✓
Watch out. A sum like 1000° would be impossible because 1000 ÷ 180 is not a whole number.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the angle sum of a hexagon, then subtract the five known angles.
- Count the sides. The shape has 6 sides, so it is a hexagon.n = 6
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.6 − 2 = 4 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.4 × 180 = 720°
- Add the five known angles. They are all 130°.5 × 130 = 650°
- Subtract from the sum. 720 − 650 = 70°
Check. 5 × 130 + 70 = 720°. ✓
Watch out. Multiply, do not add five times by hand: it is easy to slip.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Check how many triangles a hexagon is split into.
- Count the triangles. From one corner of a hexagon you can draw 3 diagonals, which makes 4 triangles (not 6).6 − 2 = 4 triangles
- Spot the mistake. Zoe multiplied by the number of sides instead of by the number of triangles.wrong: 6 × 180
- Correct the answer. 4 × 180 = 720°
Check. A hexagon has angles of 120° if regular: 6 × 120 = 720°. ✓
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 20 sides, so it is a 20-sided polygon.n = 20
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.20 − 2 = 18 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.18 × 180 = 3240°
Check. Check with the formula (n − 2) × 180 = (20 − 2) × 180 = 3240°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 10 sides, so it is a decagon.n = 10
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.10 − 2 = 8 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.8 × 180 = 1440°
Check. Check with the formula (n − 2) × 180 = (10 − 2) × 180 = 1440°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Exercise 17H Regular polygons and exterior angles
Non-calculatorKey words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular triangle has n = 3 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular triangle has 3 equal exterior angles.360 ÷ 3 = 120°
Check. The interior angle is 180 − 120 = 60°, and (3 − 2) × 180 ÷ 3 = 60° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular quadrilateral has n = 4 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular quadrilateral has 4 equal exterior angles.360 ÷ 4 = 90°
Check. The interior angle is 180 − 90 = 90°, and (4 − 2) × 180 ÷ 4 = 90° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular pentagon has n = 5 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular pentagon has 5 equal exterior angles.360 ÷ 5 = 72°
Check. The interior angle is 180 − 72 = 108°, and (5 − 2) × 180 ÷ 5 = 108° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular hexagon has n = 6 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular hexagon has 6 equal exterior angles.360 ÷ 6 = 60°
Check. The interior angle is 180 − 60 = 120°, and (6 − 2) × 180 ÷ 6 = 120° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular octagon has n = 8 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular octagon has 8 equal exterior angles.360 ÷ 8 = 45°
Check. The interior angle is 180 − 45 = 135°, and (8 − 2) × 180 ÷ 8 = 135° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular decagon has n = 10 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular decagon has 10 equal exterior angles.360 ÷ 10 = 36°
Check. The interior angle is 180 − 36 = 144°, and (10 − 2) × 180 ÷ 10 = 144° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 60°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 60°, so the number of them is 360 ÷ 60.360 ÷ 60 = 6 sides
Check. 6 × 60 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 45°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 45°, so the number of them is 360 ÷ 45.360 ÷ 45 = 8 sides
Check. 8 × 45 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 40°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 40°, so the number of them is 360 ÷ 40.360 ÷ 40 = 9 sides
Check. 9 × 40 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 30°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 30°, so the number of them is 360 ÷ 30.360 ÷ 30 = 12 sides
Check. 12 × 30 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 20°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 20°, so the number of them is 360 ÷ 20.360 ÷ 20 = 18 sides
Check. 18 × 20 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 15°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 15°, so the number of them is 360 ÷ 15.360 ÷ 15 = 24 sides
Check. 24 × 15 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 120°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 120 = 60°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 60° angles are equal, so divide 360° by the exterior angle.360 ÷ 60 = 6 sides
Check. Each interior angle of a regular hexagon is (6 − 2) × 180 ÷ 6 = 120°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 144°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 144 = 36°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 36° angles are equal, so divide 360° by the exterior angle.360 ÷ 36 = 10 sides
Check. Each interior angle of a regular decagon is (10 − 2) × 180 ÷ 10 = 144°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 150°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 150 = 30°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 30° angles are equal, so divide 360° by the exterior angle.360 ÷ 30 = 12 sides
Check. Each interior angle of a regular dodecagon is (12 − 2) × 180 ÷ 12 = 150°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 160°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 160 = 20°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 20° angles are equal, so divide 360° by the exterior angle.360 ÷ 20 = 18 sides
Check. Each interior angle of a regular 18-sided polygon is (18 − 2) × 180 ÷ 18 = 160°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 168°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 168 = 12°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 12° angles are equal, so divide 360° by the exterior angle.360 ÷ 12 = 30 sides
Check. Each interior angle of a regular 30-sided polygon is (30 − 2) × 180 ÷ 30 = 168°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular 15-sided polygon has n = 15 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(15 − 2) × 180 = 13 × 180 = 2340°
- Share the sum equally. In a regular polygon all 15 interior angles are equal, so each one is the sum divided by 15.2340 ÷ 15 = 156°
Check. Another way: each exterior angle is 360 ÷ 15 = 24°, and 180 − 24 = 156°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular octagon has n = 8 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular octagon has 8 equal exterior angles.360 ÷ 8 = 45°
Check. The interior angle is 180 − 45 = 135°, and (8 − 2) × 180 ÷ 8 = 135° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular octagon has n = 8 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(8 − 2) × 180 = 6 × 180 = 1080°
- Share the sum equally. In a regular polygon all 8 interior angles are equal, so each one is the sum divided by 8.1080 ÷ 8 = 135°
Check. Another way: each exterior angle is 360 ÷ 8 = 45°, and 180 − 45 = 135°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The number of sides is 360 divided by the exterior angle, and it must be a whole number.
- Recall the fact. The exterior angles of any polygon add up to 360°.number of sides = 360 ÷ exterior angle
- Divide. 360 ÷ 50 = 7.2
- Decide. A polygon cannot have 7.2 sides.No
Check. Exterior angles that work are factors of 360: for example 45° (8 sides) or 40° (9 sides). ✓
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the interior angle of each shape, add them, and subtract from 360° (angles around a point).
- Interior angle of the pentagon. Use (n − 2) × 180 ÷ n.3 × 180 ÷ 5 = 108°
- Interior angle of the hexagon. 4 × 180 ÷ 6 = 120°
- Add them. 108 + 120 = 228°
- Subtract from the full turn. Angles around a point add up to 360°.360 − 228 = 132°
Check. 132 + 108 + 120 = 360°. ✓
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The three interior angles must add up to 360° (angles around a point).
- Square and hexagon. Interior angles: 90° and 120°.90 + 120 = 210°
- Find the angle left. Angles around a point add up to 360°.360 − 210 = 150°
- Find n from the interior angle. The exterior angle is 180 − 150 = 30°, so n = 360 ÷ 30.n = 12
Check. A regular 12-sided polygon has interior angle 10 × 180 ÷ 12 = 150°. ✓
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of the tiles at a point must add up to exactly 360°.
- Hexagons. The interior angle is 120°.3 × 120 = 360°: they fit exactly
- Pentagons. The interior angle is 108°.3 × 108 = 324°: gap of 36°
4 × 108 = 432°: too much (overlap) - Decide. No whole number of pentagon corners makes exactly 360°.Pentagons cannot tile a floor without gaps.
Check. 360 ÷ 108 = 3.33..., not a whole number. ✓
Chapter summary
| Skill | Method |
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| x | y |
Can you…?
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Key terms
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