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GCSE Maths · EdexcelChapter 17 · Angles, polygons and symmetry
Chapter 17 · Higher tier

GCSE Maths Angles, Polygons and Symmetry (Higher): 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, Higher tier. Content references: G1, G3, G4, G6.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
Fx
17.1

Geometry words and notation

Edexcel 1MA1: G1
Grade 1-3

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Geometry words and notationComing soon

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.

line (goes on for ever)ABline segment ABpoint P
A line, a line segment AB, and a point P.

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.

parallel linesperpendicular lines
Parallel lines are marked with arrows. Perpendicular lines meet at a right angle.

An angle measures a turn. We measure it in degrees. A full turn is 360°. Angles have names that depend on their size.

acuteright angleobtusereflex
The four main kinds of angle. A straight line is 180° and a full turn is 360°.

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.

ABCangle ABC
Angle ABC is at B. You could also call it angle CBA.

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.

ABCcab
Side a is opposite vertex A, side b is opposite B, side c 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.
Worked example 17.1.1
In triangle PQR the angle at Q is marked. Write the three-letter name of the angle, and say which side is opposite it.
PQR

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.

  1. Find the vertex of the angle. The vertex is the corner where the angle is. Here the angle is marked at Q.
    Vertex = Q
  2. 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)
  3. 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).

Worked example 17.1.2
Write down the name of each type of angle: (a) 64° (b) 90° (c) 138° (d) 215°.

Key words. Acute, obtuse and reflex describe the size of an angle.

Method. Compare each angle with the benchmarks 90°, 180° and 360°.

  1. 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°.
  2. 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
  3. 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.

Worked example 17.1.3
Sketch a triangle ABC with AB = 6 cm and angle BAC = 40°, where AB = AC. Mark the equal sides and say what type of triangle it is.

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.

  1. 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)
  2. 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°
  3. 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)
  4. Name the triangle. A triangle with two equal sides is isosceles.
    Isosceles triangle
    40°6 cmABC

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-calculator
1
FWrite the correct geometry word for each description.
(a)Two lines that always stay the same distance apart.
(b)Two lines that meet at 90°.
(c)A corner of a polygon.
(d)A closed flat shape with straight sides.
(e)A polygon with all sides equal and all angles equal.
(f)The piece of a line between two points.
2
FWrite down the name of each type of angle.
(a)35°
(b)90°
(c)150°
(d)180°
(e)270°
(f)89°
3
FWrite the three-letter name of the angle marked with an arc in this triangle.
PQR
4
FTriangle ABC is shown.
ABCcab
(a)Which side is opposite angle A?
(b)Which side is opposite angle B?
(c)Which angle is opposite side c?
5
FABCD is a rectangle. Use the diagram to complete each statement with parallel or perpendicular.
ABCD
(a)AB and DC are ______.
(b)AB and BC are ______.
(c)Write down the names of two vertices that are joined by an edge of the rectangle.
6
RDraw a sketch of triangle ABC where angle ABC = 90° and AB = BC. (Mark the right angle and the equal sides.) What type of triangle is it?
7
FComplete the table for each regular polygon.
(a)A pentagon: how many sides and how many vertices?
(b)A hexagon: how many sides and how many vertices?
(c)A octagon: how many sides and how many vertices?
(d)A decagon: how many sides and how many vertices?
8
FA cube has 6 faces. Write down the number of its (a) vertices (b) edges.
(a)vertices
(b)edges
9
RSay whether each statement is true or false.
(a)A square is a type of rectangle.
(b)Parallel lines meet at a right angle.
(c)A regular hexagon has six equal sides and six equal angles.
10
FWrite the correct name for each triangle.
(a)A triangle with all three sides equal.
(b)A triangle with exactly two equal sides.
(c)A triangle with no equal sides.
(d)A triangle with one angle of 90°.
11
RTwo lines, PQ and RS, are both perpendicular to a third line, TU. What can you say about PQ and RS? Draw a sketch to explain.
12
PSHow many diagonals does a hexagon have? (A diagonal joins two vertices that are not next to each other.)
13
RShow that a polygon with n sides has n(n − 3) ÷ 2 diagonals.
17.2

Angles on a line, around a point and vertically opposite

Edexcel 1MA1: G3
Grade 1-4

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Angles on a line, around a point and vertically oppositeComing soon

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°.

ab
a + b = 180° because the angles are on a straight line.

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°.

abc
a + b + c = 360° because the angles are around a point.

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.

cab
Angles a and b are vertically opposite, so they are equal. Angle c is next to both of them on a straight line.

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.
Worked example 17.2.1
The diagram shows angles on a straight line. Work out the value of x. Give a reason for your answer.
112°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°.

  1. 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°.
  2. Add up the angles we already know. We need their total, so we can see how much of the 180° is left.
    112 = 112°
  3. 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.

Worked example 17.2.2
The diagram shows angles around a point. Work out the value of x. Give a reason for your answer.
95°120°70°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 around a point must add up to 360°. Add the angles we know and subtract from 360°.

  1. 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°.
  2. 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°
  3. 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.

Worked example 17.2.3
Two straight lines cross, as shown. Work out the size of angles a, b and c. Give a reason for each.
64°abc

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.

  1. 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°
  2. 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°
  3. 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°.

Worked example 17.2.4
The diagram shows angles on a straight line. Work out the value of x.
3x + 20°2x + 10°

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.

  1. 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°.
  2. 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°
  3. Collect like terms. Add all the x terms together, and add all the plain numbers together.
    5x + 30 = 180
  4. Get the number on the other side. Subtract 30 from both sides.
    5x = 150
  5. Divide both sides by 5. 5x means 5 × x, so dividing by 5 leaves x on its own.
    x = 150 ÷ 5 = 30
  6. 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.

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-calculator
1
FThe diagram shows angles on a straight line. Work out the value of x. Give a reason for your answer.
112°x
2
FThe diagram shows angles on a straight line. Work out the value of x.
48°x
3
FThe diagram shows angles on a straight line. Work out the value of x. Give a reason for your answer.
35°90°x
4
FThe diagram shows angles on a straight line. Work out the value of y. Give a reason for your answer.
70°y40°55°
5
FThe diagram shows angles around a point. Work out the value of a. Give a reason for your answer.
130°120°a
6
FThe diagram shows angles around a point. Work out the value of b. Give a reason for your answer.
b85°95°110°
7
FThe diagram shows angles around a point. Work out the value of c.
72°100°c64°58°
8
FTwo straight lines cross. Work out the size of each angle marked with a letter. Give a reason for each answer.
64°abc
(a)Work out angle a.
(b)Work out angle b.
(c)Work out angle c.
9
FTwo straight lines cross. Work out the size of each angle marked with a letter. Give a reason for each answer.
127°abc
(a)Work out angle a.
(b)Work out angle b.
(c)Work out angle c.
10
RTwo angles on a straight line are 107° and 83°. Karl says that the angles must be on a straight line. Is he correct? Give a reason.
11
PSWork out the size of the angle between the hands of a clock at each time. (The 12 numbers are equally spaced round the clock face.)
(a)3 o’clock
(b)5 o’clock
(c)8 o’clock (the smaller angle)
12
PSA pizza is cut into 8 equal slices. Anna takes 3 slices. What angle is left at the centre of the pizza for the remaining slices?
45°
13
FThe diagram shows angles on a straight line. Work out the size of each angle.
x°2x°
14
FThe diagram shows angles on a straight line. Work out the value of x.
3x + 20°2x + 10°
15
FThe diagram shows angles around a point. Work out the value of x.
2x°x + 30°3x − 30°
16
PSThe diagram shows angles around a point. Work out the size of the angle marked 4x + 10°.
4x + 10°3x°2x + 20°x + 10°
17
RDan has worked out the angle on a straight line next to a 112° angle. He writes 360 − 112 = 248°. Explain his mistake and find the correct angle.
18
RLines AB and CD cross at E. Angle AEC is three times angle AED. (a) Work out angle AED. (b) Write down the size of angle BEC. Give reasons.
(a)Work out angle AED.
(b)Write down the size of angle BEC.
19
RTwo straight lines cross. Prove that the angles that are opposite each other are equal. (Use a and b for two opposite angles and c for an angle next to both.)
20
RThe angles on a straight line are x, 2x and 3x + 12°. Two lines cross to make the angle 2x as one of the angles at the crossing point. Work out x and the size of the angle vertically opposite 3x + 12°.
17.3

Angles in triangles

Edexcel 1MA1: G3, G6
Grade 2-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Angles in trianglesComing soon

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.)

abcabc
The three angles of a triangle (a, b, c) fit together to make a straight line: a + b + c = 180°.

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.

xx50°ABC
AC = BC (one tick each), so the base angles at A and B are equal.

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).

50°60°xABC
The exterior angle x = 50° + 60° = 110°.

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).
Worked example 17.3.1
The diagram shows a triangle. Work out the size of angle x. Give a reason for your answer.
52°71°xABC

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°.

  1. 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°.
  2. 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°
  3. 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.

Worked example 17.3.2
The diagram shows an isosceles triangle. Work out the size of angle x. Give a reason for your answer.
48°xABC

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.

  1. 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.
  2. 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°
  3. Subtract the apex angle. This leaves the total of the two base angles.
    2x = 180 − 48 = 132°
  4. 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.

Worked example 17.3.3
The diagram shows an isosceles triangle. Work out the size of angle x. Give a reason for your answer.
38°xABC

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°.

  1. 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.
  2. Add the two base angles. We need their total to take away from 180°.
    38 + 38 = 76°
  3. 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.

Worked example 17.3.4
The side BC of triangle ABC is extended. Work out the size of angle x. Give a reason for your answer.
47°68°xABC

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.

  1. 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.
  2. 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.
  3. 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.

Worked example 17.3.5
The diagram shows a triangle. Work out the value of x. Work out the size of the largest angle.
2x + 10°3x°x + 20°ABC

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.

  1. 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°.
  2. 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°
  3. Collect like terms. Add all the x terms together, and add all the plain numbers together.
    6x + 30 = 180
  4. Get the number on the other side. Subtract 30 from both sides.
    6x = 150
  5. Divide both sides by 6. 6x means 6 × x, so dividing by 6 leaves x on its own.
    x = 150 ÷ 6 = 25
  6. 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.

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-calculator
1
FThe diagram shows a triangle. Work out the size of angle x. Give a reason for your answer.
50°65°xABC
2
FThe diagram shows a triangle. Work out the size of angle x.
x38°72°PQR
3
FThe diagram shows a triangle. Work out the size of angle x. Give a reason for your answer.
24°x105°XYZ
4
FThe diagram shows a right-angled triangle. Work out the size of angle x. Give a reason for your answer.
34°xABC
5
FThe diagram shows an isosceles triangle. Work out the size of angle x. Give a reason for your answer.
40°xABC
6
FThe diagram shows an isosceles triangle. Work out the size of angle x. Give a reason for your answer.
65°xABC
7
FThe diagram shows an isosceles triangle. Work out the size of angle x. Give a reason for your answer.
100°xPQR
8
FThe diagram shows an isosceles triangle. Work out the size of angle x. Give a reason for your answer.
35°xDEF
9
FTriangle ABC is equilateral. Work out the size of angle x. Give a reason.
xABC
10
FThe side BC of triangle ABC is extended. Work out the size of angle x. Give a reason for your answer.
50°60°xABC
11
FThe side BC of triangle ABC is extended. Work out the size of angle x. Give a reason for your answer.
35°78°xPQR
12
FTriangle ABC has angle A = 70° and angle B = 45°. Write down the type of triangle (acute-angled, right-angled or obtuse-angled). Show your working.
13
PSA roof truss is an isosceles triangle. The angle at the top is 110°. Work out the size of each of the other two angles.
14
PSThe three angles of a triangle are in the ratio 1 : 2 : 3. Work out the size of each angle and say what type of triangle it is.
15
FThe diagram shows a triangle. Work out the size of each angle.
x°2x°3x°ABC
16
PSThe diagram shows a triangle. Work out the value of x.
2x + 10°3x°x + 20°PQR
17
RExplain why a triangle cannot have two right angles.
18
RSam says: “The angles in my triangle are 70°, 60° and 60°.” Is Sam correct? Give a reason.
19
RIn triangle ABC, AC = BC and the exterior angle at C is 130°. Work out the size of angle CAB.
130°ABC
20
RThe diagram shows a triangle with one side extended. Work out the value of x.
2xx + 305x − 10ABC
21
PSThe diagram shows a triangle. Work out the size of the angle marked 4x°.
2x + 10°2x + 10°4x°ABC
22
RThe angle at the top of an isosceles triangle is x. Find an expression for a base angle in terms of x. Hence show that a base angle of an isosceles triangle is always less than 90°.
17.4

Parallel lines

Edexcel 1MA1: G3
Grade 3-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Parallel linesComing soon

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.

aa
Alternate angles (Z shape): the two angles marked a are equal.

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).

aa
Corresponding angles (F shape): the two angles marked a are equal.

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”.

ab
Co-interior angles (C or U shape): a + b = 180°.

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”.
Worked example 17.4.1
In the diagram the two horizontal lines are parallel. Work out the value of x. Give reasons for your answer.
65°x

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.

  1. 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
  2. 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.
  3. 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.

Worked example 17.4.2
In the diagram the two horizontal lines are parallel. Work out the value of y. Give reasons for your answer.
48°y

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.

  1. 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
  2. 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.
  3. 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.

Worked example 17.4.3
In the diagram the two horizontal lines are parallel. Work out the value of z. Give reasons for your answer.
70°z

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.

  1. 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
  2. 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°.
  3. 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.

Worked example 17.4.4
In the diagram the two horizontal lines are parallel. Work out the value of x. Give reasons for your answer.
128°xAB

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.

  1. 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)
  2. 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°.
  3. 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°
  4. 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.
  5. 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.

Worked example 17.4.5
In the diagram the two horizontal lines are parallel. Work out the value of x and the size of each marked angle.
3x + 34°2x + 46°

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.

  1. 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.
  2. 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.
    3x + 34 = 2x + 46
  3. Collect the x terms on one side. Subtract 2x from both sides, so that there is only one x term left.
    x + 34 = 46
  4. Collect the numbers on the other side. Subtract 34 from both sides.
    x = 12
  5. Read off the value of x.
    x = 12
  6. Substitute x = 12 to find the first angle. Now we know the value of x, put it into the expression for the angle.
    3x + 34 = 3 × 12 + 34 = 70°
  7. Substitute x = 12 to find the second angle. Now we know the value of x, put it into the expression for the angle.
    2x + 46 = 2 × 12 + 46 = 70°

Check. Check: the two angles are 70° and 70°; 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.

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-calculator
1
FIn the diagram the two horizontal lines are parallel. Work out the value of x. Give reasons for your answer.
65°x
2
FIn the diagram the two horizontal lines are parallel. Work out the value of y. Give reasons for your answer.
72°y
3
FIn the diagram the two horizontal lines are parallel. Work out the value of z. Give reasons for your answer.
65°z
4
FIn the diagram the two horizontal lines are parallel. Work out the value of a. Give reasons for your answer.
48°a
5
FIn the diagram the two horizontal lines are parallel. Work out the value of b.
133°b
6
FIn the diagram the two horizontal lines are parallel. Work out the value of c. Give reasons for your answer.
56°c
7
FIn the diagram the two horizontal lines are parallel. Work out the value of x. Give reasons for your answer.
139°x
8
FIn the diagram the two horizontal lines are parallel. Work out the value of p. Give reasons for your answer.
103°p
9
FIn the diagram the two horizontal lines are parallel. Work out the value of q. Give reasons for your answer.
104°q
10
FIn the diagram the two horizontal lines are parallel. Work out the value of m.
123°m
11
FThe diagram shows two parallel lines and a transversal, with the angles labelled a to h.
abcdefgh
(a)angle a: corresponding
(b)angle c: alternate
(c)angle d: co-interior
(d)angle d: corresponding
(e)angle d: alternate
(f)angle c: co-interior
12
PSIn the diagram of the eight angles (a to h), angle a = 62°. Work out the size of every other angle and give a reason for each.
abcdefgh
(a)angle b
(b)angle c
(c)angle d
(d)angle e
(e)angle f
(f)angle g
(g)angle h
13
RWrite true or false and give a reason.
(a)Co-interior angles are always equal.
(b)Alternate angles are equal when the lines are not parallel.
14
RAli says that angle x is 110° because co-interior angles add up to 180°. Explain his mistake and give the correct value.
70°x
15
PSThe lines are parallel. The two marked angles are (3x + 28)° and (2x + 42)°. Work out the value of x.
3x + 28°2x + 42°
16
PSThe lines are parallel. The two marked angles are (4x + 20)° and (2x + 50)°. Work out the value of x. Hence write down the size of each angle.
4x + 20°2x + 50°
17
PSThe lines are parallel. The two marked angles are (3x + 74)° and (2x + 46)°. Work out the value of x.
3x + 74°2x + 46°
18
RTwo parallel lines are crossed by a transversal. One of the angles is 90°. What can you say about all the other seven angles? Explain.

Exercise 17E  Parallel lines: multi-step angle problems and reasons

Non-calculator
1
FIn the diagram the two horizontal lines are parallel. Work out the value of x. Give reasons for your answer.
128°xAB
2
PSIn the diagram AB and CD are parallel (the arrows show this). Work out the size of angle y. Give reasons.
62°yPQ
3
PSIn the diagram, PAQ is parallel to BC. Work out (a) the size of angle x (angle PAB), (b) the size of angle BAC. Give reasons.
ABCPQx65°55°
(a)Angle PAB
(b)Angle BAC
4
PSIn the diagram, the horizontal lines through A and C are parallel. Work out the size of angle x (angle APC). Give reasons.
35°48°xAPC
5
PSIn the diagram the horizontal lines are parallel. Work out the size of angle x. Give reasons.
130°xAB
6
RThe diagram shows two parallel lines and a point P between them. Write down the size of angle APC and explain how you know.
40°55°APC
7
RABC is a triangle. The line through A is parallel to BC. Prove that angles in the triangle add up to 180°.
ABCPQacbac
8
PSIn the diagram AB and CD are parallel. Work out the size of angle x. Give reasons for your working.
52°xAB
17.5

Quadrilaterals

Edexcel 1MA1: G3, G4
Grade 2-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: QuadrilateralsComing soon

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°.

abcdABCD
A diagonal splits a quadrilateral into two triangles, so a + b + c + d = 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.

From left to right: a square, a rectangle, a parallelogram and a rhombus.
A kite, a trapezium and an isosceles trapezium.

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.

QuadrilateralSidesAnglesDiagonalsLines of symmetryOrder of rotation
Square4 equal sides; opposite sides parallelall 90°equal, bisect each other at 90°44
Rectangleopposite sides equal and parallelall 90°equal, bisect each other22
Parallelogramopposite sides equal and parallelopposite angles equalbisect each other02
Rhombus4 equal sides; opposite sides parallelopposite angles equalbisect each other at 90°; bisect the angles22
Kitetwo pairs of equal adjacent sidesone pair of equal angles (opposite each other)cross at 90°11
Isosceles trapeziumone pair of parallel sides; the other two sides equaltwo pairs of equal angles (base angles)equal11
Trapeziumexactly one pair of parallel sidesangles between the parallel sides add to 180°no special property01

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).
Worked example 17.5.1
The diagram shows a quadrilateral. Work out the size of angle x. Give a reason for your answer.
85°100°75°xABCD

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°.

  1. 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°.
  2. 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°
  3. 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.

Worked example 17.5.2
ABCD is a parallelogram. Angle A = 68°. Work out the size of angles x, y and z.
68°xyzABCD

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°.

  1. 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°)
  2. Find y. Angle C is opposite angle A.
    y = 68° (opposite angles of a parallelogram are equal)
  3. 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°.

Worked example 17.5.3
ABCD is a kite with AB = AD and CB = CD. Angle BAD = 100° and angle BCD = 60°. Work out the size of angle ABC.
100°x60°xABCD

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.

  1. 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)
  2. 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
  3. Collect and subtract. 100 + 60 = 160, and the two equal angles are what is left.
    2x = 360 − 160 = 200
  4. 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.

Worked example 17.5.4
ABCD is an isosceles trapezium with AB parallel to DC and AD = BC. Angle DAB = 65°. Work out the size of angle ADC.
65°xABCD

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°.

  1. 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
  2. State the fact. Co-interior angles add up to 180°.
    65 + x = 180
  3. 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.

Worked example 17.5.5
The diagram shows a quadrilateral. Work out the value of x.
2x + 10°3x°x + 20°4x − 10°ABCD

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.

  1. 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°.
  2. 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°
  3. Collect like terms. Add all the x terms together, and add all the plain numbers together.
    10x + 20 = 360
  4. Get the number on the other side. Subtract 20 from both sides.
    10x = 340
  5. Divide both sides by 10. 10x means 10 × x, so dividing by 10 leaves x on its own.
    x = 340 ÷ 10 = 34
  6. 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.

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-calculator
1
FThe diagram shows a quadrilateral. Work out the size of angle x. Give a reason for your answer.
85°100°75°xABCD
2
FThe diagram shows a quadrilateral. Work out the size of angle x. Give a reason for your answer.
x110°90°70°PQRS
3
FThe diagram shows a quadrilateral. Work out the size of angle x. Give a reason for your answer.
125°xABCD
4
FThe diagram shows a quadrilateral. Work out the size of angle y.
65°y130°82°WXYZ
5
FWrite the name of the quadrilateral that matches each description. Give the most specific name.
(a)Four equal sides and four right angles.
(b)Two pairs of equal sides, and opposite sides parallel, and no right angles.
(c)Four equal sides, with no right angles.
(d)Exactly one pair of parallel sides.
(e)Two pairs of equal sides next to each other, with one line of symmetry.
(f)Four right angles and opposite sides equal, but sides not all equal.
6
FABCD is a parallelogram. Angle A = 68°. Work out the size of each angle marked with a letter. Give a reason for each answer.
68°xyzABCD
(a)Work out angle x (angle B).
(b)Work out angle y (angle C).
(c)Work out angle z (angle D).
7
FABCD is a parallelogram. Angle A = 125°. Work out the size of each angle marked with a letter. Give a reason for each answer.
125°xyzABCD
(a)Work out angle x (angle B).
(b)Work out angle y (angle C).
(c)Work out angle z (angle D).
8
FABCD is a kite. Angle BAD = 110° and angle BCD = 50°. Work out angle ABC. Give reasons.
110°x50°xABCD
9
PSABCD is a trapezium with AB parallel to DC. Work out the size of (a) angle x (b) angle y. Give reasons.
75°70°xyABCD
(a)Angle x (angle BCD)
(b)Angle y (angle ADC)
10
RABCD is a rhombus. Angle ABC = 110°. The diagonal AC is drawn. Work out the size of angle BAC. Give reasons.
110°ABCD
11
PSABCD is a rectangle and AC is a diagonal. Angle CAB = 37°. Work out the size of angle ACB. Give reasons.
37°xABCD
12
FWrite down the number of lines of symmetry of each quadrilateral.
(a)A square: how many lines of symmetry?
(b)A rectangle: how many lines of symmetry?
(c)A parallelogram: how many lines of symmetry?
(d)A rhombus: how many lines of symmetry?
(e)A kite: how many lines of symmetry?
(f)A isosceles trapezium: how many lines of symmetry?
13
RWrite true or false. Give a reason for each answer.
(a)Every rectangle is a square.
(b)A rhombus is a parallelogram.
(c)A trapezium has two pairs of parallel sides.
14
FThe diagram shows a quadrilateral. Work out the value of x.
x°2x°3x°4x°ABCD
15
PSThe diagram shows a quadrilateral. Work out the size of the angle marked 4x − 10°.
2x + 10°3x°x + 20°4x − 10°PQRS
16
RThree angles of a quadrilateral are 90°, 90° and 90°. Write down the fourth angle. What can you say about the quadrilateral? Give a reason.
17
RShow that the opposite angles of a parallelogram are equal.
abcdABCD
18
RThe angles of a quadrilateral are x, (x + 30)°, 2x and (3x − 20)°. Show that the quadrilateral has an angle of 100°.
17.6

Interior and exterior angles of polygons

Edexcel 1MA1: G3
Grade 3-7

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Interior and exterior angles of polygonsComing soon

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°

A hexagon splits into 4 triangles (4 × 180° = 720°). A pentagon splits into 3 triangles (3 × 180° = 540°).
PolygonSides nTriangles n − 2Angle sum
Triangle31180°
Quadrilateral42360°
Pentagon53540°
Hexagon64720°
Heptagon75900°
Octagon861080°
Nonagon971260°
Decagon1081440°

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.

120°60°135°45°
Regular hexagon: interior angle 120°, exterior angle 60°. Regular octagon: interior 135°, exterior 45°.

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.
Worked example 17.6.1
Work out the sum of the interior angles of an octagon.

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°.

  1. Count the sides. The shape has 8 sides, so it is a octagon.
    n = 8
  2. 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
  3. 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.

Worked example 17.6.2
Work out the size of each interior angle of a regular nonagon (a 9-sided polygon).

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.

  1. 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
  2. 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°
  3. 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.

Worked example 17.6.3
The diagram shows a pentagon. Work out the size of angle x. Give a reason for your answer.
100°120°90°110°xABCDE

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.

  1. Count the sides. The shape has 5 sides, so it is a pentagon.
    n = 5
  2. 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
  3. Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.
    3 × 180 = 540°
  4. Add the angles we know. We need their total to take from the sum.
    100 + 120 + 90 + 110 = 420°
  5. 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).

Worked example 17.6.4
Each exterior angle of a regular polygon is 24°. How many sides does the polygon have?

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.

  1. Write down what we know. The polygon is regular, so all its exterior angles are equal.
    each exterior angle = 24°
  2. Recall the fact. The number of exterior angles equals the number of sides.
    The exterior angles of any polygon add up to 360°.
  3. 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).

Worked example 17.6.5
Each interior angle of a regular polygon is 156°. How many sides does it have?

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.

  1. 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 = 156°
  2. Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.
    180 − 156 = 24°
  3. Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 24° angles are equal, so divide 360° by the exterior angle.
    360 ÷ 24 = 15 sides

Check. Each interior angle of a regular 15-sided polygon is (15 − 2) × 180 ÷ 15 = 156°. ✓

Watch out. Do not divide the interior angle into 360 directly: use the exterior 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-calculator
1
FWork out the sum of the interior angles of each polygon.
(a)a pentagon
(b)a hexagon
(c)a octagon
(d)a decagon
(e)a dodecagon
2
FThe diagram shows a pentagon. Work out the size of angle x. Give a reason for your answer.
100°120°90°110°xABCDE
3
FThe diagram shows a hexagon. Work out the size of angle y. Give a reason for your answer.
130°100°115°125°95°yPQRSTU
4
PSThe diagram shows a heptagon. Work out the size of angle z. Give a reason for your answer.
150°110°120°140°125°135°z
5
FThe diagram shows a pentagon. Work out the size of angle a. Give a reason for your answer.
128°a98°112°102°
6
PSThe diagram shows a hexagon. Work out the size of angle b.
105°120°135°95°140°b
7
PSThe sum of the interior angles of a polygon is 1260°. How many sides does the polygon have?
8
FWork out the size of each interior angle of these regular polygons.
(a)a regular pentagon
(b)a regular hexagon
(c)a regular octagon
(d)a regular decagon
(e)a regular dodecagon
9
RIs it possible for the interior angles of a polygon to add up to 900°? Give a reason. If so, how many sides does it have?
10
PSA hexagon has five interior angles of 130°. Work out the sixth angle.
11
PSThe interior angles of a pentagon are x°, 2x°, (x + 30)°, (2x − 20)° and (3x − 10)°. Work out the value of x.
12
RShow that the sum of the interior angles of any polygon with n sides is (n − 2) × 180°.
13
RZoe says: “A hexagon has 6 sides, so the sum of its interior angles is 6 × 180 = 1080°.” Explain her mistake and give the correct answer.

Exercise 17H  Regular polygons and exterior angles

Non-calculator
1
FWork out the size of each exterior angle of these regular polygons.
(a)a regular triangle
(b)a regular quadrilateral
(c)a regular pentagon
(d)a regular hexagon
(e)a regular octagon
(f)a regular decagon
2
FEach exterior angle of a regular polygon is given. Work out the number of sides.
(a)60°
(b)45°
(c)40°
(d)30°
(e)20°
(f)15°
3
PSWork out the number of sides of a regular polygon with each interior angle given.
(a)120°
(b)144°
(c)150°
(d)160°
(e)168°
4
FWork out the size of the interior angle of a regular polygon with 15 sides.
5
PSA stop sign is a regular octagon. Work out (a) the size of each exterior angle (b) the size of each interior angle.
135°45°
(a)each exterior angle
(b)each interior angle
6
RCan a regular polygon have an exterior angle of 50°? Give a reason.
7
PSA regular pentagon and a regular hexagon are placed with one corner at a point so that they do not overlap. Work out the angle that is left to complete a full turn.
8
RThree regular polygons meet at a point so that there are no gaps: a square, a regular hexagon and a regular n-sided polygon. Work out n.
9
RWhy can you tile a floor with regular hexagons but not with regular pentagons? (Tiles must meet at a point with no gaps and no overlaps.)
10
RThe exterior angle of a regular polygon is one fifth of its interior angle. Work out the number of sides.
11
REach interior angle of a regular polygon is (4x + 10)° and each exterior angle is (x − 5)°. Work out x and the number of sides.
12
RProve that the exterior angles of a triangle add up to 360°. (Use the three interior angles a, b and c.)

Chapter summary

SkillMethod
xy

Can you…?

  • x

Key terms

x
y
← Chapter 16: Units and compound measuresChapter 18: Perimeter, area and circles →

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