GCSE Maths: Geometry words and notation (Higher)
Part of the GCSE Maths angles, polygons and symmetry chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Geometry words and notation
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. From each vertex, count the diagonals, then correct for counting every diagonal twice.
- Count diagonals from one vertex. A vertex cannot be joined to itself or to its two neighbours, because those are sides. The other 6 − 3 = 3 vertices give diagonals.6 − 3 = 3
- Do this for all 6 vertices. Each of the 6 vertices has 3 diagonals.6 × 3 = 18
- Divide by 2. Every diagonal has two ends, so it has been counted twice.18 ÷ 2 = 9
Check. Use the formula n(n − 3) ÷ 2 with n = 6: 6 × 3 ÷ 2 = 9. ✓
Watch out. Do not forget to divide by 2: a diagonal from A to D is the same as one from D to A.
Method. Count the diagonals from one vertex, multiply by the number of vertices, then halve.
- Diagonals from one vertex. A vertex is joined by an edge to its two neighbours and cannot be joined to itself, so it is joined by diagonals to the other n − 3 vertices.n − 3 diagonals from each vertex
- Do this for every vertex. There are n vertices.n × (n − 3)
- Allow for double counting. Each diagonal joins two vertices, so it appears twice in that total.n(n − 3) ÷ 2
Check. Test with a pentagon (n = 5): 5 × 2 ÷ 2 = 5 diagonals, which is the five lines of a pentagram. ✓
Watch out. A triangle has no diagonals: n = 3 gives 3 × 0 ÷ 2 = 0.