Level 3 · Mathematics M3.5

Angles, triangles and symmetry

Work out missing angles, draw accurate triangles, and find symmetry in shapes.

Find missing angles on a line, round a point and between parallel lines. Show that a triangle has 180° and a quadrilateral 360°, and find the angles of any polygon. Draw triangles with ruler and compasses, cut an angle in half, find line symmetry and turning symmetry, and draw solids, nets and views.

You are ready for this book if

You can measure an angle with a protractor to the nearest degree.

Written and drawn by attnom. Free to read, print and share.

M3.5.1 Chapter 1

Angles on a line and round a point

Two sticks cross on the ground. One angle is 40°. Can I find the other three without measuring?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Stand still and turn all the way round. That is a full turn. A full turn is 360°. Now turn only half way. That is a half turn, and it is 180°.

Two lines can meet at one point. We call that point the vertex. Angles that share a vertex can fill a straight line or a full turn.

Draw it first

Angles on a straight line

  1. A straight line with a point O on it. A second line leaves O and makes two angles with the first. The angle on the right is 60° and the angle on the left is x, which is 120°. Under the picture it says 60° + 120° = 180°.O

    1 Draw a straight line. Mark a point O on it.

  2. A straight line with a point O on it. A second line leaves O and makes two angles with the first. The angle on the right is 60° and the angle on the left is x, which is 120°. Under the picture it says 60° + 120° = 180°.O

    2 Draw a second line from O. It makes two angles.

  3. A straight line with a point O on it. A second line leaves O and makes two angles with the first. The angle on the right is 60° and the angle on the left is x, which is 120°. Under the picture it says 60° + 120° = 180°.O60°x60° + 120° = 180°

    3 The two angles fill the straight line. They add to 180°.

This is the first rule. Angles on a line add to 180°. If you know one, take it away from 180° to find the other.

Worked example

Two angles fill a straight line. One angle is 118°. What is the other angle, x?

  1. Angles on a straight line add to 180°.
  2. So 118° + x = 180°.
  3. x = 180° − 118° = 62°.

Answer: 62°

A straight line with a point O. A second line from O makes an angle of 118° on the left and an angle x on the right. The angle x is 62°.Ox118°
The angle x is 62°. A protractor on this page would read 62°.

Now go all the way round the vertex. Angles round a point fill a full turn, so they add to 360°.

Worked example

Four angles meet at a point. Three of them are 95°, 120° and 70°. What is the fourth angle, x?

  1. Angles round a point add to 360°.
  2. The three known angles add to 95° + 120° + 70° = 285°.
  3. x = 360° − 285° = 75°.

Answer: 75°

Four angles round a point. They are 95°, 120°, 70° and x. The angle x is 75°.95°120°70°x
The four angles fill a full turn: 95° + 120° + 70° + 75° = 360°.

Two straight lines can cross. They make four angles. Angles that face each other across the vertex are vertically opposite angles, or opposite angles for short. They are equal.

Worked example

Two straight lines cross. One angle is 38°. What are the four angles?

  1. Call the angles a, b, c and d. Angle a is 38°.
  2. Angle b and angle a fill a straight line, so b = 180° − 38° = 142°.
  3. Angle c is opposite a, so c = 38°.
  4. Angle d is opposite b, so d = 142°.
  5. The four angles are 38°, 142°, 38° and 142°.

Answer: 38°, 142°, 38° and 142°

Two straight lines cross. The angles going round are a, b, c and d. Angles a and c are 38° and are shaded with hatching. Angles b and d are 142°.abcd
The shaded angles a and c are opposite and equal. Angles b and d are opposite and equal.

See it three ways

With things

Lay a long stick on the ground. Lean a short stick on it. The short stick makes two angles with the long stick, one on each side.

As a picture

A long line with a point O and a leaning line from O. The angle on the right is a, which is 50°. The angle on the left is b, which is 130°.Oab

In symbols

a + b = 180°. When a is 50°, b is 180° − 50° = 130°.

Try it

Cross two lines and measure

You need: a sheet of paper, a ruler, a protractor, a pencil.

  1. Draw two straight lines that cross.
  2. Name the four angles a, b, c and d, going round the vertex.
  3. Measure each angle with the protractor.
  4. Add a and b. Then add b and c.

What you will see: Opposite angles are the same size. Angles side by side add to 180°. A degree or two of difference is normal when you measure.

If you cannot try it, this is what happens.
Two crossing lines with the four angles measured: a is 62°, b is 118°, c is 62° and d is 118°.62°118°62°118°

A wrong idea many people have

Any two angles that touch add up to 180°.

Why it feels right: On a straight line, angles that touch do add up to 180°.

The test that breaks it: Look at the corner of a page. It is 90°. Draw a line from the corner into the page, for example at 30°. You get two angles, 30° and 60°. They touch, but they add to 90°.

What is true: Angles that touch add to 180° only when together they fill a straight line.

Key words

vertex
the point where two lines meet.
angles on a line
angles that fill a straight line. They add to 180°.
angles round a point
angles that fill a full turn. They add to 360°.
vertically opposite angles
angles that face each other where two straight lines cross. They are equal. We also call them opposite angles.

Back to your guess

The answer is B: Yes. The opposite angle is 40° and the other two are 140°.

Angles next to each other on a straight line add to 180°. So 180° − 40° = 140°. The angle opposite 40° is also 40°. The four angles are 40°, 140°, 40° and 140°.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Two angles fill a straight line. One is 70°. What is the angle x?

    A straight line with a point on it and a leaning line. The angle on the right is 70°. The angle on the left is x.70°x
    Tap your answer.

    1. Yes, that is right. The two angles fill a straight line, so they add to 180°. 180° − 70° = 110°.

    2. Not this one. You took 70° from 360°. A straight line is a half turn, 180°, not 360°. Go to M3.5.1. Try another.

    3. Not this one. You took 70° from 90°. Angles on a straight line add to 180°, not 90°. Go to M2.5.4. Try another.

    See the answer

    110°

    The two angles fill a straight line, so they add to 180°. 180° − 70° = 110°.

    • If you chose B: You took 70° from 360°. A straight line is a half turn, 180°, not 360°. Go to M3.5.1.
    • If you chose C: You took 70° from 90°. Angles on a straight line add to 180°, not 90°. Go to M2.5.4.

  2. Three angles fill a straight line. Two of them are 45° and 65°. What is the angle x?

    A straight line with a point on it and two leaning lines. The angles from the right are 45°, 65° and x.45°65°x
    Tap your answer.

    1. Not this one. That is 45° + 65°, the two angles you know. Take it away from 180° to find x. Go to M3.5.1. Try another.

    2. Not this one. You took only 45° from 180°. Take away 65° as well. Go to M3.5.1. Try another.

    3. Yes, that is right. The known angles add to 45° + 65° = 110°. Then 180° − 110° = 70°.

    See the answer

    70°

    The known angles add to 45° + 65° = 110°. Then 180° − 110° = 70°.

    • If you chose A: That is 45° + 65°, the two angles you know. Take it away from 180° to find x. Go to M3.5.1.
    • If you chose B: You took only 45° from 180°. Take away 65° as well. Go to M3.5.1.

  3. Three angles fill a full turn round a point. Two of them are 80° and 60°. What is the angle x?

    Three angles round a point. They are 80°, 60° and x. The angle x is the big angle that goes round the rest.80°60°x
    Tap your answer.

    1. Not this one. That is 80° + 60°, the two angles you know. Take it away from 360° to find x. Go to M3.5.1. Try another.

    2. Yes, that is right. Angles round a point add to 360°. The known angles add to 80° + 60° = 140°. Then 360° − 140° = 220°.

    3. Not this one. You used 180°, as for a straight line. Angles round a point fill a full turn, which is 360°. Go to M3.5.1. Try another.

    See the answer

    220°

    Angles round a point add to 360°. The known angles add to 80° + 60° = 140°. Then 360° − 140° = 220°.

    • If you chose A: That is 80° + 60°, the two angles you know. Take it away from 360° to find x. Go to M3.5.1.
    • If you chose C: You used 180°, as for a straight line. Angles round a point fill a full turn, which is 360°. Go to M3.5.1.

  4. Two straight lines cross. One angle is 52°. What is the angle c, opposite it?

    Two straight lines cross. Going round, the angles are 52°, b, c and d. The angle c is opposite the 52° angle.52°bcd
    Tap your answer.

    1. Yes, that is right. Angle c is opposite the 52° angle, and opposite angles are equal. So c = 52°.

    2. Not this one. That is the angle next to 52°, on the same straight line. The opposite angle faces 52° across the vertex. Go to M3.5.1. Try another.

    3. Not this one. You took 52° from 90°. Opposite angles are equal, so c is the same size as 52°. Go to M3.5.1. Try another.

    See the answer

    52°

    Angle c is opposite the 52° angle, and opposite angles are equal. So c = 52°.

    • If you chose B: That is the angle next to 52°, on the same straight line. The opposite angle faces 52° across the vertex. Go to M3.5.1.
    • If you chose C: You took 52° from 90°. Opposite angles are equal, so c is the same size as 52°. Go to M3.5.1.

  5. Which pair of angles is always equal?

    Two straight lines cross. Hatching marks the two angles that face each other across the vertex.
    Tap your answer.

    1. Not this one. Side by side on a straight line they add to 180°. They are equal only when both are 90°. Go to M3.5.1. Try another.

    2. Not this one. Angles round a point add to 360°. They are not always equal. Go to M3.5.1. Try another.

    3. Yes, that is right. Two straight lines that cross make two pairs of opposite angles. Each pair is equal.

    See the answer

    Opposite angles where two straight lines cross

    Two straight lines that cross make two pairs of opposite angles. Each pair is equal.

    • If you chose A: Side by side on a straight line they add to 180°. They are equal only when both are 90°. Go to M3.5.1.
    • If you chose B: Angles round a point add to 360°. They are not always equal. Go to M3.5.1.

  6. Two angles on a straight line are 117° and x. What is x?

    See the answer

    63°

    180° − 117° = 63°.

  7. Three angles round a point are 85°, 140° and 60°. A fourth angle x completes the turn. What is x?

    See the answer

    75°

    85° + 140° + 60° = 285°. Then 360° − 285° = 75°.

Keep it

Copy this into your exercise book.

Angles on a straight line add to 180°. Angles round a point add to 360°. Where two straight lines cross, opposite angles are equal.

Draw: Two crossing lines. Mark the four angles and write 40°, 140°, 40° and 140° on them.

M3.5.2 Chapter 2

Angles and parallel lines

A road crosses two parallel railway tracks. I know one angle. Can I find the other seven?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Parallel lines run side by side. They stay the same distance apart and never meet. We mark them with arrows that point the same way.

A line that crosses two parallel lines is a transversal. It leans the same way at both crossings. So the angles at the top crossing match the angles at the bottom crossing.

Draw it first

The same place at each crossing

  1. Two parallel lines, each with an arrow. A line crosses both of them. At the top crossing the angle on the upper right is 65° and is shaded with hatching. At the bottom crossing the angle in the same place is x and is also shaded. Under the picture it says x = 65°.

    1 Draw two parallel lines. Mark each with an arrow.

  2. Two parallel lines, each with an arrow. A line crosses both of them. At the top crossing the angle on the upper right is 65° and is shaded with hatching. At the bottom crossing the angle in the same place is x and is also shaded. Under the picture it says x = 65°.

    2 Draw a line across both of them.

  3. Two parallel lines, each with an arrow. A line crosses both of them. At the top crossing the angle on the upper right is 65° and is shaded with hatching. At the bottom crossing the angle in the same place is x and is also shaded. Under the picture it says x = 65°.65°xx = 65°

    3 The angle in the same place at each crossing is the same size.

The two shaded angles are in the same place at each crossing. They are corresponding angles. Look for the letter F. Corresponding angles between parallel lines are equal.

Two parallel lines with arrows and a crossing line. Shaded with hatching are the angle at the upper left of the top crossing and the angle at the upper left of the bottom crossing. Both are 115°.115°115°
These corresponding angles make an F shape. Both are 115°.

Some angles sit between the parallel lines, on opposite sides of the transversal. They are alternate angles. Look for the letter Z. Alternate angles between parallel lines are equal.

Worked example

The two lines are parallel. One angle is 72°. What is the alternate angle x?

  1. Both angles are between the parallel lines.
  2. They are on opposite sides of the transversal.
  3. They make a Z shape, so they are alternate angles.
  4. Alternate angles are equal, so x = 72°.

Answer: 72°

Two parallel lines with arrows and a crossing line. Between the lines, the angle at the lower left of the top crossing is 72° and the angle at the upper right of the bottom crossing is x. Both are shaded with hatching. They make a Z shape.72°x
The angle x is 72°. The two angles lie in a Z shape.

Worked example

The two lines are parallel. The angle at the top is 65°. What is the angle y at the bottom?

  1. The angle next to 65° on the top line is 180° − 65° = 115°.
  2. The angle y is in the same place at the bottom crossing.
  3. So y is a corresponding angle, and y = 115°.

Answer: 115°

Two parallel lines with arrows and a crossing line. The angle at the upper right of the top crossing is 65°. The angle at the upper left of the bottom crossing is y.65°y
The angle y is 115°. It sits next to the angle that corresponds to 65°, so y = 180° − 65°.

See it three ways

With things

Draw a line along each long edge of a ruler. The two lines are parallel. Draw a third line across both.

As a picture

Two parallel lines with arrows and a crossing line. The angle at the upper right of the top crossing is 70°. The angle in the same place at the bottom crossing is also 70°. Both are shaded with hatching.70°70°

In symbols

The angle at the top = the angle in the same place at the bottom. 70° = 70°.

Try it

Measure the angles between parallel lines

You need: a sheet of paper, a ruler, a protractor, a pencil.

  1. Lay the ruler on the paper. Draw a line along each long edge.
  2. Draw a straight line across both lines.
  3. Measure the four angles at the top crossing. Then measure the four at the bottom.
  4. Compare the angles that are in the same place at each crossing.

What you will see: Each angle matches the one in the same place at the other crossing. Only two sizes appear. A degree or two of difference is normal.

If you cannot try it, this is what happens.
Two parallel lines with arrows and a crossing line. At each crossing the four angles are marked. The upper right and lower left angles are 65°. The other two are 115°. The angles in the same place at the two crossings match.65°115°65°115°65°115°65°115°

A wrong idea many people have

All eight angles are equal.

Why it feels right: The lines are parallel, so everything looks the same.

The test that breaks it: Measure two angles side by side at one crossing. One is small and one is big. They fill a straight line, so they add to 180°.

What is true: Only two sizes appear. Corresponding angles are equal and alternate angles are equal. Angles side by side add to 180°.

Key words

parallel lines
straight lines that stay the same distance apart and never meet.
transversal
a straight line that crosses two or more other lines.
corresponding angles
angles in the same place at each crossing, in an F shape. Between parallel lines they are equal.
alternate angles
angles between the lines on opposite sides of the transversal, in a Z shape. Between parallel lines they are equal.

Back to your guess

The answer is B: Yes. Each angle is the one I know, or 180° minus it.

The tracks are parallel, so the road meets both of them at the same slant. Only two sizes appear. Four angles are the one you know. The other four are 180° minus it.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The lines are parallel. The angle x is in the same place as the 70° angle. What is x?

    Two parallel lines with arrows and a crossing line. The angle at the upper right of the top crossing is 70°. The angle in the same place at the bottom crossing is x.70°x
    Tap your answer.

    1. Yes, that is right. The angle x is in the same place at the other crossing. Corresponding angles between parallel lines are equal, so x = 70°.

    2. Not this one. That is 180° − 70°, the angle next to it. The angle in the same place is a corresponding angle, so it is equal to 70°. Go to M3.5.2. Try another.

    3. Not this one. You took 70° from 90°. Corresponding angles between parallel lines are equal. Go to M3.5.2. Try another.

    See the answer

    70°

    The angle x is in the same place at the other crossing. Corresponding angles between parallel lines are equal, so x = 70°.

    • If you chose B: That is 180° − 70°, the angle next to it. The angle in the same place is a corresponding angle, so it is equal to 70°. Go to M3.5.2.
    • If you chose C: You took 70° from 90°. Corresponding angles between parallel lines are equal. Go to M3.5.2.

  2. The lines are parallel. What are the two shaded angles called?

    Two parallel lines with arrows and a crossing line. Two angles between the lines are shaded with hatching. One is at the lower left of the top crossing and one is at the upper right of the bottom crossing. They make a Z shape.
    Tap your answer.

    1. Not this one. Corresponding angles are in the same place at each crossing, like an F. These are between the lines on opposite sides, like a Z. Go to M3.5.2. Try another.

    2. Not this one. Opposite angles face each other at one crossing. These two are at different crossings. Go to M3.5.1. Try another.

    3. Yes, that is right. Both angles are between the parallel lines, on opposite sides of the transversal. They make a Z shape, so they are alternate angles.

    See the answer

    Alternate angles

    Both angles are between the parallel lines, on opposite sides of the transversal. They make a Z shape, so they are alternate angles.

    • If you chose A: Corresponding angles are in the same place at each crossing, like an F. These are between the lines on opposite sides, like a Z. Go to M3.5.2.
    • If you chose B: Opposite angles face each other at one crossing. These two are at different crossings. Go to M3.5.1.

  3. The lines are parallel. One angle is 125°. The angle y is its alternate angle. What is y?

    Two parallel lines with arrows and a crossing line. Between the lines, the angle at the lower right of the top crossing is 125° and the angle at the upper left of the bottom crossing is y. Both are shaded with hatching.125°y
    Tap your answer.

    1. Not this one. That is the angle next to 125° on a straight line. The alternate angle is equal to 125°. Go to M3.5.2. Try another.

    2. Yes, that is right. The two angles make a Z shape between the parallel lines. Alternate angles are equal, so y = 125°.

    3. Not this one. You took 125° from 360°. Alternate angles are equal, so y is 125°. Go to M3.5.2. Try another.

    See the answer

    125°

    The two angles make a Z shape between the parallel lines. Alternate angles are equal, so y = 125°.

    • If you chose A: That is the angle next to 125° on a straight line. The alternate angle is equal to 125°. Go to M3.5.2.
    • If you chose C: You took 125° from 360°. Alternate angles are equal, so y is 125°. Go to M3.5.2.

  4. The lines are parallel. One angle is 64°. What is the angle x?

    Two parallel lines with arrows and a crossing line. The angle at the upper right of the top crossing is 64°. The angle at the lower right of the bottom crossing is x.64°x
    Tap your answer.

    1. Yes, that is right. At the top, the lower right angle is next to 64° on a straight line. So it is 180° − 64° = 116°. The angle x is in the same place at the bottom, so x = 116°.

    2. Not this one. The angle x is not in the same place as 64°. It is next to the corresponding angle, so it fills a straight line with it. Go to M3.5.2. Try another.

    3. Not this one. You took 64° from 90°. Find the angle in the same place as x at the top, then use 180°. Go to M3.5.1. Try another.

    See the answer

    116°

    At the top, the lower right angle is next to 64° on a straight line. So it is 180° − 64° = 116°. The angle x is in the same place at the bottom, so x = 116°.

    • If you chose B: The angle x is not in the same place as 64°. It is next to the corresponding angle, so it fills a straight line with it. Go to M3.5.2.
    • If you chose C: You took 64° from 90°. Find the angle in the same place as x at the top, then use 180°. Go to M3.5.1.

  5. Which sentence about two parallel lines and a transversal is true?

    Tap your answer.

    1. Not this one. Alternate angles are equal. The angles that add to 180° are side by side on a straight line. Go to M3.5.2. Try another.

    2. Not this one. Only two sizes appear. Angles side by side add to 180°, so they are equal only when both are 90°. Go to M3.5.2. Try another.

    3. Yes, that is right. Corresponding angles are in the same place at each crossing. Between parallel lines they are equal.

    See the answer

    Corresponding angles are equal.

    Corresponding angles are in the same place at each crossing. Between parallel lines they are equal.

    • If you chose A: Alternate angles are equal. The angles that add to 180° are side by side on a straight line. Go to M3.5.2.
    • If you chose B: Only two sizes appear. Angles side by side add to 180°, so they are equal only when both are 90°. Go to M3.5.2.

  6. The lines are parallel. An angle at the top is 58°. The angle in the same place at the bottom is x. What is the angle next to x on the straight line?

    See the answer

    122°

    The corresponding angle x is 58°. The angle next to x fills a straight line with it, so it is 180° − 58° = 122°.

  7. From an earlier chapter

    Three angles fill a straight line. Two of them are 38° and 52°. What is the third angle?

    See the answer

    90°

    38° + 52° = 90°. Then 180° − 90° = 90°.

Keep it

Copy this into your exercise book.

A line across parallel lines makes two sizes of angle. Corresponding angles (F) are equal. Alternate angles (Z) are equal. Angles side by side add to 180°.

Draw: Two parallel lines with arrows and a line across them. Shade a pair of corresponding angles and a pair of alternate angles.

M3.5.3 Chapter 3

Triangles

Do the three angles of every triangle add up to the same number?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

A triangle has three corners and three angles. Add the three angles of any triangle. The answer is always 180°. This is the angle sum of a triangle.

Draw it first

Tear off the corners

  1. A triangle with angles 70°, 50° and 60°. Below it the three angles sit side by side at one point on a straight line. From the right they are 60°, then 70°, then 50°, each on the side where it began. Under the picture it says 60° + 70° + 50° = 180°.70°50°60°

    1 Draw a triangle. Its angles are 70°, 50° and 60°.

  2. A triangle with angles 70°, 50° and 60°. Below it the three angles sit side by side at one point on a straight line. From the right they are 60°, then 70°, then 50°, each on the side where it began. Under the picture it says 60° + 70° + 50° = 180°.70°50°60°60°70°50°

    2 Tear off the three corners. Put them side by side with their points together.

  3. A triangle with angles 70°, 50° and 60°. Below it the three angles sit side by side at one point on a straight line. From the right they are 60°, then 70°, then 50°, each on the side where it began. Under the picture it says 60° + 70° + 50° = 180°.70°50°60°60°70°50°60° + 70° + 50° = 180°

    3 They fill a straight line. So the three angles add to 180°.

Chapter 2 shows why. Draw a line through the top corner, parallel to the base. The alternate angles match the two bottom corners.

The same works for every triangle. So you can find a missing angle. Add the two angles you know, and take the answer from 180°.

Worked example

A triangle has angles of 65° and 48°. What is the third angle, x?

  1. The three angles add to 180°.
  2. The two known angles add to 65° + 48° = 113°.
  3. x = 180° − 113° = 67°.

Answer: 67°

A triangle with corners A, B and C. The angle at A is x, which is 67°. The angle at B is 65° and the angle at C is 48°.ABCx65°48°
The angle x is 67°: 67° + 65° + 48° = 180°.

Triangles have names. A name tells you how many sides are equal. Small tick marks on sides show which sides are equal.

Four triangles with their names: equilateral, isosceles, scalene and right angled. The equilateral triangle has one tick on each of its three equal sides and one arc on each of its three equal angles of 60°. The isosceles triangle has one tick on each of its two equal sides and one arc on each of its two equal angles of 70°. The scalene triangle has no equal sides and no ticks. The right angled triangle has a small square at its right angle of 90°.equilateralisoscelesscaleneright angled
Ticks show equal sides. Matching arcs show equal angles. The little square shows a right angle.

An equilateral triangle has three equal sides and three angles of 60°. An isosceles triangle has two equal sides and two equal angles. The equal angles are opposite the equal sides.

A scalene triangle has no equal sides and no equal angles. A right angled triangle has one angle of 90°. A right angled triangle can also be isosceles.

Worked example

An isosceles triangle has an angle of 40° between its two equal sides. What are the other two angles?

  1. The three angles add to 180°.
  2. The other two angles add to 180° − 40° = 140°.
  3. The two equal sides make two equal angles.
  4. Each of them is 140° ÷ 2 = 70°.

Answer: 70° and 70°

An isosceles triangle with corners A, B and C. Tick marks show that sides AB and AC are equal. The angle at A is 40°. The angles at B and C are both 70°.ABC40°70°70°
The angles at B and C are equal because the sides AB and AC are equal.

Worked example

A right angled triangle has an angle of 35°. What is the third angle?

  1. One angle is 90°.
  2. The three angles add to 180°.
  3. The other two angles add to 180° − 90° = 90°.
  4. One of them is 35°.
  5. The third angle is 90° − 35° = 55°.

Answer: 55°

A right angled triangle with corners A, B and C. A small square marks the right angle at A. The angle at B is 35° and the angle at C is x, which is 55°.ABC35°x
The two angles that are not the right angle add to 90°.

See it three ways

With things

Draw a triangle on paper. Tear off its three corners. Put them side by side with their points together.

As a picture

Three angles side by side at one point on a straight line. From the right they are 60°, 70° and 50°.60°70°50°

In symbols

a + b + c = 180°. For example 70° + 50° + 60° = 180°.

Try it

Tear the corners off a triangle

You need: a sheet of paper, a pencil.

  1. Draw a triangle on the paper. Make it any shape you like.
  2. Mark each corner with a small curved line.
  3. Tear off the three corners.
  4. Put the corners side by side with their points together.

What you will see: The three corners fill a straight line. That is 180°. It works for every triangle you try.

If you cannot try it, this is what happens.
Three torn corners side by side at one point on a straight line. From the right they are 60°, 70° and 50°.60°70°50°

A wrong idea many people have

A bigger triangle has a bigger angle sum.

Why it feels right: A big triangle has longer sides, so it seems to have more of everything.

The test that breaks it: Draw a small triangle and a big triangle. Measure the angles of each and add them up. Both add to about 180°.

What is true: Every triangle has an angle sum of 180°. Longer sides do not make the angles add to more.

Key words

angle sum
the total of the angles in a shape. For a triangle it is 180°.
equilateral triangle
a triangle with three equal sides and three angles of 60°.
isosceles triangle
a triangle with two equal sides and two equal angles.
scalene triangle
a triangle with no equal sides.
right angled triangle
a triangle with one angle of 90°.

Back to your guess

The answer is A: Yes. They always add up to 180°.

Tear off the three corners of a paper triangle and put them side by side. They fill a straight line, which is 180°. This works for every triangle, big or small.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A triangle has angles of 55° and 72°. What is the third angle x?

    A triangle with corners A, B and C. The angle at A is x. The angle at B is 55° and the angle at C is 72°.ABCx55°72°
    Tap your answer.

    1. Not this one. That is 55° + 72°, the two angles you know. Take it away from 180° to find x. Go to M3.5.3. Try another.

    2. Yes, that is right. 55° + 72° = 127°. Then 180° − 127° = 53°.

    3. Not this one. You took 127° from 360°. The angles of a triangle add to 180°, not 360°. Go to M3.5.3. Try another.

    See the answer

    53°

    55° + 72° = 127°. Then 180° − 127° = 53°.

    • If you chose A: That is 55° + 72°, the two angles you know. Take it away from 180° to find x. Go to M3.5.3.
    • If you chose C: You took 127° from 360°. The angles of a triangle add to 180°, not 360°. Go to M3.5.3.

  2. This isosceles triangle has an angle of 50° between its equal sides. What is the angle x?

    An isosceles triangle with corners A, B and C. Ticks show that AB and AC are equal. The angle at A is 50°. The angles at B and C are both x.ABC50°xx
    Tap your answer.

    1. Not this one. That is 180° − 50°. The two equal angles share it, so each is half of 130°. Go to M3.5.3. Try another.

    2. Not this one. The angle of 50° is at the top. The two base angles are equal to each other, but they are not 50°. Go to M3.5.3. Try another.

    3. Yes, that is right. The other two angles add to 180° − 50° = 130°. They are equal, so each is 130° ÷ 2 = 65°.

    See the answer

    65°

    The other two angles add to 180° − 50° = 130°. They are equal, so each is 130° ÷ 2 = 65°.

    • If you chose A: That is 180° − 50°. The two equal angles share it, so each is half of 130°. Go to M3.5.3.
    • If you chose B: The angle of 50° is at the top. The two base angles are equal to each other, but they are not 50°. Go to M3.5.3.

  3. The sides of this triangle are 4 cm, 5 cm and 6 cm. What kind of triangle is it?

    A triangle with corners A, B and C. Side BC is 5 cm, side CA is 4 cm and side AB is 6 cm. No two sides are equal.ABC5 cm4 cm6 cm
    Tap your answer.

    1. Yes, that is right. All three sides have a different length, so the triangle is scalene.

    2. Not this one. An isosceles triangle has two sides of the same length. These three sides are all different. Go to M3.5.3. Try another.

    3. Not this one. An equilateral triangle has three equal sides. These sides are 4 cm, 5 cm and 6 cm. Go to M3.5.3. Try another.

    See the answer

    Scalene triangle

    All three sides have a different length, so the triangle is scalene.

    • If you chose B: An isosceles triangle has two sides of the same length. These three sides are all different. Go to M3.5.3.
    • If you chose C: An equilateral triangle has three equal sides. These sides are 4 cm, 5 cm and 6 cm. Go to M3.5.3.

  4. A triangle has a right angle and an angle of 38°. What is the third angle?

    A right angled triangle with corners A, B and C. A small square marks the right angle at A. The angle at B is 38°. The angle at C is x.ABC38°x
    Tap your answer.

    1. Not this one. You took only 38° from 180°. Take away the right angle, 90°, as well. Go to M3.5.3. Try another.

    2. Yes, that is right. The three angles add to 180°. 90° + 38° = 128°. Then 180° − 128° = 52°.

    3. Not this one. The triangle already has one angle of 90°. The other two add to 90°, so the third is smaller. Go to M3.5.3. Try another.

    See the answer

    52°

    The three angles add to 180°. 90° + 38° = 128°. Then 180° − 128° = 52°.

    • If you chose A: You took only 38° from 180°. Take away the right angle, 90°, as well. Go to M3.5.3.
    • If you chose C: The triangle already has one angle of 90°. The other two add to 90°, so the third is smaller. Go to M3.5.3.

  5. Which three angles can be the angles of a triangle?

    Tap your answer.

    1. Not this one. These add to 190°. The angles of a triangle add to 180°. Go to M3.5.3. Try another.

    2. Not this one. These add to 190°. The angles of a triangle add to 180°. Go to M3.5.3. Try another.

    3. Yes, that is right. 90° + 50° + 40° = 180°. The other sets add to 190°.

    See the answer

    90°, 50° and 40°

    90° + 50° + 40° = 180°. The other sets add to 190°.

    • If you chose A: These add to 190°. The angles of a triangle add to 180°. Go to M3.5.3.
    • If you chose B: These add to 190°. The angles of a triangle add to 180°. Go to M3.5.3.

  6. Two angles of a triangle are 48° and 77°. What is the third angle?

    See the answer

    55°

    48° + 77° = 125°. Then 180° − 125° = 55°.

  7. From an earlier chapter

    Two lines are parallel. An angle at the top is 70°. The angle y is next to the corresponding angle at the bottom, on a straight line. What is y?

    See the answer

    110°

    The corresponding angle at the bottom is 70°. The angle y fills a straight line with it, so y = 180° − 70° = 110°.

Keep it

Copy this into your exercise book.

The three angles of a triangle add to 180°. Equilateral: 3 equal sides, all angles 60°. Isosceles: 2 equal sides, 2 equal angles. Scalene: none equal. Right angled: one 90° angle.

Draw: A triangle with its three corners drawn as angles side by side on a straight line. Write 180° under it.

M3.5.4 Chapter 4

Angles in polygons

Do the four angles of any four sided shape add up to the same number?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

A quadrilateral is a shape with four straight sides. A square, a rectangle and a kite are quadrilaterals. So are many shapes with no name.

A line that joins two corners that are not next to each other is a diagonal. A diagonal cuts a quadrilateral into two triangles.

Draw it first

Cut a quadrilateral into two triangles

  1. A four sided shape with corners that measure 80°, 100°, 70° and 110°. A dashed diagonal from the bottom left corner to the top right corner cuts it into two triangles, numbered 1 and 2. Under the picture it says 2 triangles × 180° = 360°.

    1 Draw a quadrilateral.

  2. A four sided shape with corners that measure 80°, 100°, 70° and 110°. A dashed diagonal from the bottom left corner to the top right corner cuts it into two triangles, numbered 1 and 2. Under the picture it says 2 triangles × 180° = 360°.12

    2 Draw one diagonal. It makes two triangles.

  3. A four sided shape with corners that measure 80°, 100°, 70° and 110°. A dashed diagonal from the bottom left corner to the top right corner cuts it into two triangles, numbered 1 and 2. Under the picture it says 2 triangles × 180° = 360°.80°100°70°110°2 triangles × 180° = 360°

    3 Each triangle has 180°, so the four angles add to 2 × 180° = 360°.

So the angles of every quadrilateral add to 360°. You can use this to find a missing angle.

Worked example

Three angles of a quadrilateral are 80°, 100° and 70°. What is the fourth angle, x?

  1. The four angles of a quadrilateral add to 360°.
  2. The three known angles add to 80° + 100° + 70° = 250°.
  3. x = 360° − 250° = 110°.

Answer: 110°

A four sided shape. Going round from the bottom left, the angles are 80°, 100°, 70° and x. The angle x is 110°.80°100°70°x
The four angles add to 80° + 100° + 70° + 110° = 360°.

A polygon is a flat shape with straight sides. Choose one corner. Draw diagonals from it to every corner it can reach. You get triangles.

A pentagon with five sides. Dashed diagonals from the top corner cut it into three triangles, numbered 1, 2 and 3.123
A pentagon has 5 sides and makes 3 triangles.
A hexagon with six sides. Dashed diagonals from one corner cut it into four triangles, numbered 1, 2, 3 and 4.1234
A hexagon has 6 sides and makes 4 triangles.

The number of triangles is always 2 less than the number of sides. So a polygon with n sides makes n − 2 triangles. Its angle sum is (n − 2) × 180°.

A table with three columns: sides, triangles and sum. It has rows for 3, 4, 5 and 6 sides. They make 1, 2, 3 and 4 triangles. The angle sums are 180°, 360°, 540° and 720°.sidestrianglessum31180°42360°53540°64720°
Every extra side adds one more triangle, and 180° more.

Worked example

What is the angle sum of a polygon with 8 sides?

  1. An 8 sided polygon makes 8 − 2 = 6 triangles.
  2. Each triangle has an angle sum of 180°.
  3. The angle sum is 6 × 180° = 1 080°.

Answer: 1 080°

An octagon with eight sides. Dashed diagonals from one corner cut it into six triangles, numbered 1 to 6.123456
An octagon makes 6 triangles. 6 × 180° = 1 080°.

A regular polygon has equal sides and equal angles. To find one angle, share the angle sum between all the corners.

Worked example

What is each angle of a regular hexagon?

  1. A hexagon has 6 sides.
  2. It makes 6 − 2 = 4 triangles.
  3. The angle sum is 4 × 180° = 720°.
  4. A regular hexagon has 6 equal angles.
  5. Each angle is 720° ÷ 6 = 120°.

Answer: 120°

A regular hexagon. All six sides are equal and each of the six angles is 120°.120°120°120°120°120°120°
Six angles of 120° make 720°.

See it three ways

With things

Draw a four sided shape on paper. Draw one diagonal. Cut along it. You have two triangles.

As a picture

A four sided shape cut by a dashed diagonal into two triangles, numbered 1 and 2.12

In symbols

2 triangles × 180° = 360°. So the four angles add to 360°.

Try it

Tear the corners off a quadrilateral

You need: a sheet of paper, a pencil.

  1. Draw a four sided shape on the paper. Make it any shape you like.
  2. Mark each corner with a small curved line.
  3. Tear off the four corners.
  4. Put the corners together with their points touching.

What you will see: The four corners fit round a point. That is a full turn, 360°. This works for every four sided shape.

If you cannot try it, this is what happens.
Four torn corners round a point. They are 80°, 100°, 70° and 110°. Together they fill a full turn.80°100°70°110°

A wrong idea many people have

The angles of a quadrilateral add to 180°, like a triangle.

Why it feels right: A triangle is the first shape we learn, and its angle sum is 180°.

The test that breaks it: Look at a rectangle. Its four angles are all 90°. Four angles of 90° add to 360°, not 180°.

What is true: A quadrilateral is two triangles joined along a diagonal. So its angle sum is 2 × 180° = 360°.

Key words

quadrilateral
a shape with four straight sides. Its angles add to 360°.
diagonal
a line that joins two corners that are not next to each other.
polygon
a flat shape with straight sides. With n sides, its angles add to (n − 2) × 180°.
regular polygon
a polygon with all sides equal and all angles equal.

Back to your guess

The answer is A: Yes. They always add up to 360°.

Draw a line from one corner to the opposite corner. It cuts the shape into two triangles. Each triangle has 180°, so the four angles add up to 2 × 180° = 360°.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Three angles of a quadrilateral are 100°, 90° and 60°. What is the fourth angle x?

    A four sided shape. Going round from the bottom left, the angles are 100°, 90°, 60° and x.100°90°60°x
    Tap your answer.

    1. Not this one. That is 100° + 90° + 60°, the three angles you know. Take it away from 360° to find x. Go to M3.5.4. Try another.

    2. Yes, that is right. 100° + 90° + 60° = 250°. Then 360° − 250° = 110°.

    3. Not this one. You may have used 180°, the angle sum of a triangle. The angles of a quadrilateral add to 360°. Go to M3.5.4. Try another.

    See the answer

    110°

    100° + 90° + 60° = 250°. Then 360° − 250° = 110°.

    • If you chose A: That is 100° + 90° + 60°, the three angles you know. Take it away from 360° to find x. Go to M3.5.4.
    • If you chose C: You may have used 180°, the angle sum of a triangle. The angles of a quadrilateral add to 360°. Go to M3.5.4.

  2. What is the angle sum of a pentagon?

    A pentagon with five sides. Dashed diagonals from one corner cut it into three triangles, numbered 1, 2 and 3.123
    Tap your answer.

    1. Not this one. You multiplied 5 sides by 180°. Take 2 away first. A pentagon makes 3 triangles, not 5. Go to M3.5.4. Try another.

    2. Not this one. That is the angle sum of a quadrilateral. A pentagon has one more side, so one more triangle. Go to M3.5.4. Try another.

    3. Yes, that is right. A pentagon makes 5 − 2 = 3 triangles. 3 × 180° = 540°.

    See the answer

    540°

    A pentagon makes 5 − 2 = 3 triangles. 3 × 180° = 540°.

    • If you chose A: You multiplied 5 sides by 180°. Take 2 away first. A pentagon makes 3 triangles, not 5. Go to M3.5.4.
    • If you chose B: That is the angle sum of a quadrilateral. A pentagon has one more side, so one more triangle. Go to M3.5.4.

  3. This polygon is a regular pentagon. What is the angle x?

    A regular pentagon with five equal sides. One of its angles is marked x.x
    Tap your answer.

    1. Yes, that is right. The angle sum is 540°. The 5 angles are equal, so each is 540° ÷ 5 = 108°.

    2. Not this one. That is the angle sum of all five angles. Share it between the 5 angles. Go to M3.5.4. Try another.

    3. Not this one. That is 360° ÷ 5. Use the angle sum of a pentagon, 540°, and share it between the 5 angles. Go to M3.5.4. Try another.

    See the answer

    108°

    The angle sum is 540°. The 5 angles are equal, so each is 540° ÷ 5 = 108°.

    • If you chose B: That is the angle sum of all five angles. Share it between the 5 angles. Go to M3.5.4.
    • If you chose C: That is 360° ÷ 5. Use the angle sum of a pentagon, 540°, and share it between the 5 angles. Go to M3.5.4.

  4. Draw all the diagonals from one corner of a polygon with 7 sides. How many triangles do you get?

    Tap your answer.

    1. Not this one. That is the number of sides. The corner cannot join itself or the 2 corners next to it. It draws 4 diagonals and makes 5 triangles. Go to M3.5.4. Try another.

    2. Yes, that is right. A polygon with n sides makes n − 2 triangles. 7 − 2 = 5.

    3. Not this one. You took away only 1. Take away 2: a polygon with n sides makes n − 2 triangles. Go to M3.5.4. Try another.

    See the answer

    5

    A polygon with n sides makes n − 2 triangles. 7 − 2 = 5.

    • If you chose A: That is the number of sides. The corner cannot join itself or the 2 corners next to it. It draws 4 diagonals and makes 5 triangles. Go to M3.5.4.
    • If you chose C: You took away only 1. Take away 2: a polygon with n sides makes n − 2 triangles. Go to M3.5.4.

  5. What is each angle of a regular hexagon?

    A regular hexagon with six equal sides, cut by dashed diagonals from one corner into four triangles, numbered 1 to 4.1234
    Tap your answer.

    1. Not this one. That is the sum of all six angles. Share it between the 6 corners. Go to M3.5.4. Try another.

    2. Not this one. 360° ÷ 6 is the turn at the centre, not the inside angle of the hexagon. Go to M3.5.4. Try another.

    3. Yes, that is right. A hexagon makes 4 triangles, so its angle sum is 4 × 180° = 720°. The 6 angles are equal, so each is 720° ÷ 6 = 120°.

    See the answer

    120°

    A hexagon makes 4 triangles, so its angle sum is 4 × 180° = 720°. The 6 angles are equal, so each is 720° ÷ 6 = 120°.

    • If you chose A: That is the sum of all six angles. Share it between the 6 corners. Go to M3.5.4.
    • If you chose B: 360° ÷ 6 is the turn at the centre, not the inside angle of the hexagon. Go to M3.5.4.

  6. What is the angle sum of a polygon with 9 sides?

    See the answer

    1 260°

    9 − 2 = 7 triangles. 7 × 180° = 1 260°.

  7. From an earlier chapter

    An isosceles triangle has an angle of 30° between its equal sides. What is each of the other two angles?

    See the answer

    75°

    The other two angles add to 180° − 30° = 150°. They are equal, so each is 150° ÷ 2 = 75°.

Keep it

Copy this into your exercise book.

A polygon with n sides makes n − 2 triangles. Its angle sum is (n − 2) × 180°. A quadrilateral adds to 360°. Each angle of a regular hexagon is 120°.

Draw: A hexagon cut into four triangles by diagonals from one corner. Write 4 × 180° = 720° beside it.

M3.5.5 Chapter 5

Drawing with ruler and compasses

I know the three sides of a triangle. Can I draw it exactly with a ruler and compasses?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

A construction is a drawing made with a ruler and compasses. Compasses draw circles. A piece of a circle is an arc. You leave every arc on the page, so anyone can see how you drew it.

The pictures in this chapter are smaller than real size. Every length in one picture uses the same scale. Draw your own at full size.

Draw it first

A triangle from three sides

  1. Drawing a triangle with sides of 5 cm, 4 cm and 3 cm. Step 1: a base from B to C, 5 cm long. Step 2: an arc about B with the compasses set to 3 cm. Step 3: an arc about C with the compasses set to 4 cm. The arcs cross at A. Step 4: lines join A to B and A to C. The labels are B, C, A, 5 cm, 3 cm and 4 cm.BC5 cm

    1 Draw the base BC, 5 cm long, with a ruler.

  2. Drawing a triangle with sides of 5 cm, 4 cm and 3 cm. Step 1: a base from B to C, 5 cm long. Step 2: an arc about B with the compasses set to 3 cm. Step 3: an arc about C with the compasses set to 4 cm. The arcs cross at A. Step 4: lines join A to B and A to C. The labels are B, C, A, 5 cm, 3 cm and 4 cm.BC5 cm3 cm

    2 Set the compasses to 3 cm. Put the point on B and draw an arc.

  3. Drawing a triangle with sides of 5 cm, 4 cm and 3 cm. Step 1: a base from B to C, 5 cm long. Step 2: an arc about B with the compasses set to 3 cm. Step 3: an arc about C with the compasses set to 4 cm. The arcs cross at A. Step 4: lines join A to B and A to C. The labels are B, C, A, 5 cm, 3 cm and 4 cm.BC5 cm3 cm4 cmA

    3 Set the compasses to 4 cm. Put the point on C and draw an arc that crosses the first. Mark the crossing A.

  4. Drawing a triangle with sides of 5 cm, 4 cm and 3 cm. Step 1: a base from B to C, 5 cm long. Step 2: an arc about B with the compasses set to 3 cm. Step 3: an arc about C with the compasses set to 4 cm. The arcs cross at A. Step 4: lines join A to B and A to C. The labels are B, C, A, 5 cm, 3 cm and 4 cm.BC5 cm3 cm4 cmA

    4 Join A to B and A to C with the ruler. Leave the arcs in.

Worked example

Draw a triangle with sides of 6 cm, 5 cm and 4 cm.

  1. Draw the longest side, 6 cm, as the base BC.
  2. Set the compasses to 4 cm. Put the point on B and draw an arc.
  3. Set the compasses to 5 cm. Put the point on C and draw an arc that crosses the first.
  4. Mark the crossing A. Join A to B and A to C.
  5. Check with the ruler. AB is 4 cm and AC is 5 cm.

Answer: A triangle with sides of 6 cm, 5 cm and 4 cm

A triangle ABC drawn with compasses. The base BC is 6 cm. Side AB is 4 cm and side CA is 5 cm. The two arcs that cross at A are left in the picture.BC6 cm4 cm5 cmA
The arcs stay in. They show the compasses were set to 4 cm and 5 cm.

Do the arcs always meet? No. The two shorter sides together must be longer than the longest side. If they are not, the arcs miss each other.

A base BC of 7 cm. An arc about B set to 3 cm and an arc about C set to 2 cm. The arcs do not meet, so there is no third corner. The label says the arcs do not meet.BC7 cm3 cm2 cmthe arcs do not meet
Sides of 7 cm, 3 cm and 2 cm make no triangle. 3 + 2 is less than 7.

Draw it first

A triangle from two sides and the angle between them

  1. Drawing a triangle with a base of 6 cm, an angle of 50° at B and a side of 4 cm. Step 1: the base from B to C, 6 cm long. Step 2: a line from B at 50° to the base. Step 3: an arc about B with the compasses set to 4 cm. It crosses the line at A. Step 4: A is joined to C. The labels are B, C, A, 6 cm, 50° and 4 cm.BC6 cm

    1 Draw the base BC, 6 cm long.

  2. Drawing a triangle with a base of 6 cm, an angle of 50° at B and a side of 4 cm. Step 1: the base from B to C, 6 cm long. Step 2: a line from B at 50° to the base. Step 3: an arc about B with the compasses set to 4 cm. It crosses the line at A. Step 4: A is joined to C. The labels are B, C, A, 6 cm, 50° and 4 cm.BC6 cm50°

    2 Put the protractor on B. Mark 50° from the base and draw a line through the mark.

  3. Drawing a triangle with a base of 6 cm, an angle of 50° at B and a side of 4 cm. Step 1: the base from B to C, 6 cm long. Step 2: a line from B at 50° to the base. Step 3: an arc about B with the compasses set to 4 cm. It crosses the line at A. Step 4: A is joined to C. The labels are B, C, A, 6 cm, 50° and 4 cm.BC6 cm50°4 cmA

    3 Set the compasses to 4 cm. Put the point on B and draw an arc across the line. Mark the crossing A.

  4. Drawing a triangle with a base of 6 cm, an angle of 50° at B and a side of 4 cm. Step 1: the base from B to C, 6 cm long. Step 2: a line from B at 50° to the base. Step 3: an arc about B with the compasses set to 4 cm. It crosses the line at A. Step 4: A is joined to C. The labels are B, C, A, 6 cm, 50° and 4 cm.BC6 cm50°4 cmA

    4 Join A to C.

Worked example

Draw a triangle. Two sides are 6 cm and 4 cm. The angle between them is 50°.

  1. Draw the base BC, 6 cm long.
  2. At B, use a protractor to mark 50°. Draw a line through the mark.
  3. Set the compasses to 4 cm. Put the point on B and draw an arc across the line.
  4. Call the crossing A. Join A to C.
  5. The angle at B is the angle between the 6 cm side and the 4 cm side.

Answer: A triangle with sides of 6 cm and 4 cm and an angle of 50°

A triangle ABC. The base BC is 6 cm. The angle at B is 50°. Side BA is 4 cm. The arc about B that crosses the line at A is left in.BC6 cm50°4 cmA
The angle at B is the angle between the two sides you were given.

Compasses can also cut an angle in half. We say they bisect the angle. The line that does this is the angle bisector.

Draw it first

Cut an angle of 80° in half

  1. An angle of 80° with its corner at O. Step 1: the angle with its two arms. Step 2: an arc about O that crosses the arms at P and Q. Step 3: two arcs of the same size, one about P and one about Q, that cross at R. Step 4: a line from O through R cuts the angle into two halves of 40°. The labels are O, P, Q, R, 80° and 40° twice.O80°

    1 Draw an angle of 80°. Its corner is O.

  2. An angle of 80° with its corner at O. Step 1: the angle with its two arms. Step 2: an arc about O that crosses the arms at P and Q. Step 3: two arcs of the same size, one about P and one about Q, that cross at R. Step 4: a line from O through R cuts the angle into two halves of 40°. The labels are O, P, Q, R, 80° and 40° twice.OPQ

    2 Put the compass point on O. Draw an arc that crosses both arms, at P and at Q.

  3. An angle of 80° with its corner at O. Step 1: the angle with its two arms. Step 2: an arc about O that crosses the arms at P and Q. Step 3: two arcs of the same size, one about P and one about Q, that cross at R. Step 4: a line from O through R cuts the angle into two halves of 40°. The labels are O, P, Q, R, 80° and 40° twice.OPQR

    3 Keep the compasses the same. Draw an arc from P and an arc from Q. They cross at R.

  4. An angle of 80° with its corner at O. Step 1: the angle with its two arms. Step 2: an arc about O that crosses the arms at P and Q. Step 3: two arcs of the same size, one about P and one about Q, that cross at R. Step 4: a line from O through R cuts the angle into two halves of 40°. The labels are O, P, Q, R, 80° and 40° twice.OPQR40°40°

    4 Draw a line from O through R. It cuts the angle into two halves of 40°.

See it three ways

With things

Take two strings, one 3 cm and one 4 cm. Hold their ends on the two ends of a 5 cm line. Pull the loose ends together. They meet at one point.

As a picture

A base BC of 5 cm. An arc about B set to 3 cm and an arc about C set to 4 cm cross at A. Lines join A to B and A to C.BC5 cm3 cm4 cmA

In symbols

3 + 4 is more than 5, so the strings meet. 2 + 3 is less than 7. Strings of 2 cm and 3 cm cannot meet over a 7 cm line.

Try it

Draw a triangle with compasses

You need: a sheet of paper, a ruler, a pair of compasses, a pencil.

Be safe: Compasses have a sharp point. Hold them by the top. Keep the point away from your face and from other people. Close them and cover the point when you put them down.

  1. Draw a line 7 cm long. Call its ends B and C.
  2. Set the compasses to 5 cm. Put the point on B and draw an arc above the line.
  3. Set the compasses to 6 cm. Put the point on C and draw an arc that crosses the first.
  4. Mark the crossing A. Join A to B and A to C. Measure AB and AC with the ruler.

What you will see: AB is 5 cm and AC is 6 cm. Anyone who draws sides of 7 cm, 6 cm and 5 cm gets the same shape.

If you cannot try it, this is what happens.
A triangle ABC with a base BC of 7 cm, side CA of 6 cm and side AB of 5 cm. The two arcs that cross at A are left in.BC7 cm5 cm6 cmA

A wrong idea many people have

You can draw a triangle with any three lengths.

Why it feels right: Any three sticks seem as if they should make a triangle.

The test that breaks it: Try sides of 2 cm, 3 cm and 7 cm. Draw the 7 cm base and the two arcs. They never meet.

What is true: The two shorter sides together must be longer than the longest side.

Key words

construction
a drawing made with a ruler and compasses, with the arcs left in.
compasses
a tool that draws circles and arcs. It has a sharp point.
arc
a piece of a circle.
bisect
to cut exactly in half. The line that bisects an angle is the angle bisector.

Back to your guess

The answer is A: Yes. Two arcs show where the third corner goes.

Draw one side as the base. Set the compasses to the second side and draw an arc from one end. Set them to the third side and draw an arc from the other end. The arcs cross at the third corner.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. You draw a triangle with sides of 5 cm, 3 cm and 4 cm. The base BC is 5 cm. You draw an arc from B with the compasses set to 3 cm. What do you set them to for the arc from C?

    A base BC of 5 cm with an arc about B set to 3 cm already drawn. The arc about C is not drawn yet. The labels are B, C, 5 cm and 3 cm.BC5 cm3 cm
    Tap your answer.

    1. Not this one. That is the length of the base. The arc from C uses the third side. Go to M3.5.5. Try another.

    2. Not this one. That is the setting you already used for the arc from B. The arc from C uses the side that is left. Go to M3.5.5. Try another.

    3. Yes, that is right. The base uses 5 cm and the arc from B uses 3 cm. The third side is 4 cm, so set the compasses to 4 cm.

    See the answer

    4 cm

    The base uses 5 cm and the arc from B uses 3 cm. The third side is 4 cm, so set the compasses to 4 cm.

    • If you chose A: That is the length of the base. The arc from C uses the third side. Go to M3.5.5.
    • If you chose B: That is the setting you already used for the arc from B. The arc from C uses the side that is left. Go to M3.5.5.

  2. Can you draw a triangle with sides of 7 cm, 3 cm and 2 cm?

    A base BC of 7 cm. An arc about B set to 3 cm and an arc about C set to 2 cm. The arcs do not meet. The label says the arcs do not meet.BC7 cm3 cm2 cmthe arcs do not meet
    Tap your answer.

    1. Not this one. Not any three. The two short sides must be longer together than the long side. Here 3 + 2 is less than 7. Go to M3.5.5. Try another.

    2. Yes, that is right. 3 cm and 2 cm together are 5 cm, which is less than 7 cm. The arcs cannot cross.

    3. Not this one. The ruler is not the problem. The sides are too short to reach each other, whatever tool you use. Go to M3.5.5. Try another.

    See the answer

    No. The arcs do not meet.

    3 cm and 2 cm together are 5 cm, which is less than 7 cm. The arcs cannot cross.

    • If you chose A: Not any three. The two short sides must be longer together than the long side. Here 3 + 2 is less than 7. Go to M3.5.5.
    • If you chose C: The ruler is not the problem. The sides are too short to reach each other, whatever tool you use. Go to M3.5.5.

  3. Which three lengths can be the sides of a triangle?

    Tap your answer.

    1. Yes, that is right. 5 + 6 = 11, which is more than 10. The other two sets have two sides that add to 7 or 4, not more than the third.

    2. Not this one. 2 + 2 is exactly 4. The arcs only touch on the base, so you get a flat line, not a triangle. Go to M3.5.5. Try another.

    3. Not this one. 3 + 4 is only 7, which is less than 8. The arcs would not meet. Go to M3.5.5. Try another.

    See the answer

    5 cm, 6 cm, 10 cm

    5 + 6 = 11, which is more than 10. The other two sets have two sides that add to 7 or 4, not more than the third.

    • If you chose B: 2 + 2 is exactly 4. The arcs only touch on the base, so you get a flat line, not a triangle. Go to M3.5.5.
    • If you chose C: 3 + 4 is only 7, which is less than 8. The arcs would not meet. Go to M3.5.5.

  4. A triangle has a base BC of 6 cm and a side BA of 4 cm. The angle at B is 50°. How do you find the corner A?

    A base BC of 6 cm. A dashed line leaves B at an angle of 50°. The corner A is not marked yet. The labels are B, C, 6 cm and 50°.BC6 cm50°
    Tap your answer.

    1. Not this one. The 6 cm is the base BC, which is already drawn. The arc uses the side BA. Go to M3.5.5. Try another.

    2. Not this one. That draws a triangle from three sides. Here you know an angle, so you draw a line at 50° from B first. Go to M3.5.5. Try another.

    3. Yes, that is right. The corner A is on the 50° line, and 4 cm from B. The arc about B at 4 cm crosses the line at A.

    See the answer

    Draw a line at 50° from B. Set the compasses to 4 cm and draw an arc about B across the line.

    The corner A is on the 50° line, and 4 cm from B. The arc about B at 4 cm crosses the line at A.

    • If you chose A: The 6 cm is the base BC, which is already drawn. The arc uses the side BA. Go to M3.5.5.
    • If you chose B: That draws a triangle from three sides. Here you know an angle, so you draw a line at 50° from B first. Go to M3.5.5.

  5. The arcs are drawn from P and from Q and they cross at R. The line from O through R cuts the angle in half. What is each half?

    An angle of 100° with its corner at O. An arc about O crosses the arms at P and Q. Two arcs about P and Q cross at R. The line from O through R is not drawn yet. The labels are O, P, Q, R and 100°.O100°PQR
    Tap your answer.

    1. Not this one. That is double the angle. Cutting an angle in half makes the parts smaller, not bigger. Go to M3.5.5. Try another.

    2. Yes, that is right. The line through R cuts the angle exactly in half. 100° ÷ 2 = 50°.

    3. Not this one. That is a quarter of 100°. Cutting in half means dividing by 2. Go to M3.5.5. Try another.

    See the answer

    50°

    The line through R cuts the angle exactly in half. 100° ÷ 2 = 50°.

    • If you chose A: That is double the angle. Cutting an angle in half makes the parts smaller, not bigger. Go to M3.5.5.
    • If you chose C: That is a quarter of 100°. Cutting in half means dividing by 2. Go to M3.5.5.

  6. An angle of 84° is cut in half with compasses. How big is each half?

    See the answer

    42°

    Each half is 84° ÷ 2 = 42°.

  7. From an earlier chapter

    A right angled triangle has an angle of 28°. What is the third angle?

    See the answer

    62°

    The three angles add to 180°. 90° + 28° = 118°. Then 180° − 118° = 62°.

Keep it

Copy this into your exercise book.

Three sides: draw the base, then two arcs that cross at the third corner. Two sides and an angle: mark the angle, then use an arc for the length. Arcs from P and Q cut an angle in half.

Draw: A base BC with two arcs crossing at A. Join A to B and A to C. Write the three side lengths in cm.

M3.5.6 Chapter 6

Line symmetry and reflection

In how many ways can I fold a rectangle so that both halves match?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

A line of symmetry cuts a shape into two halves that match. Fold the shape along the line. The two halves lie exactly on top of each other.

Draw it first

Test for a line of symmetry

  1. An isosceles triangle. Step 1: the triangle. Step 2: a dashed line from the top corner to the middle of the base, numbered 1. Step 3: the words fold along line 1. The two halves match.

    1 Draw the shape on paper.

  2. An isosceles triangle. Step 1: the triangle. Step 2: a dashed line from the top corner to the middle of the base, numbered 1. Step 3: the words fold along line 1. The two halves match.1

    2 Draw a line you think is a line of symmetry.

  3. An isosceles triangle. Step 1: the triangle. Step 2: a dashed line from the top corner to the middle of the base, numbered 1. Step 3: the words fold along line 1. The two halves match.1fold along line 1

    3 Fold along the line. If the two halves match exactly, it is a line of symmetry.

A shape can have no line of symmetry, one line, or many. We number each line so none is counted twice.

Six shapes with every line of symmetry dashed and numbered. The rectangle has 2 lines, the square 4, the parallelogram none, the kite 1, the pentagon 5 and the house 1.12rectangle1234squareparallelogram1kite12345pentagon1house
The parallelogram has no line of symmetry. Its halves match only when you turn one over.

Worked example

How many lines of symmetry does an equilateral triangle have?

  1. Each corner has a line from it to the middle of the opposite side.
  2. Fold along that line. The two halves match.
  3. The triangle has 3 corners, so it has 3 lines.
  4. An equilateral triangle has 3 lines of symmetry.

Answer: 3

An equilateral triangle with its three lines of symmetry dashed and numbered 1, 2 and 3. Each line goes from a corner to the middle of the opposite side.123
Three lines, one from each corner.

A line of symmetry works like a mirror. The shape on one side is the reflection of the shape on the other. The reflection is called the image.

Draw it first

Draw a reflection

  1. A grid with a dashed mirror line and a triangle ABC to its left. Step 2: dashed lines go from A, B and C straight across the mirror line. Step 3: the image A', B' and C' is drawn the same distance from the mirror line on the other side. The grid is numbered along the bottom and the left side.0123456789100123456mirror lineABC

    1 Draw the shape and the mirror line on a grid.

  2. A grid with a dashed mirror line and a triangle ABC to its left. Step 2: dashed lines go from A, B and C straight across the mirror line. Step 3: the image A', B' and C' is drawn the same distance from the mirror line on the other side. The grid is numbered along the bottom and the left side.0123456789100123456mirror lineABC

    2 From each corner, draw a line straight across the mirror line.

  3. A grid with a dashed mirror line and a triangle ABC to its left. Step 2: dashed lines go from A, B and C straight across the mirror line. Step 3: the image A', B' and C' is drawn the same distance from the mirror line on the other side. The grid is numbered along the bottom and the left side.0123456789100123456mirror lineABCA'B'C'

    3 Mark each new corner the same distance from the mirror line. Join them to make the image.

Each corner of the image is as far from the mirror line as its matching corner. The image is the same size and shape.

A point such as (2, 3) means 2 across and 3 up. The mirror line x = 5 is the upright line through 5 on the bottom scale. The line y = 3 lies flat through 3 on the side scale.

Worked example

Reflect the triangle with corners A(1, 1), B(4, 1) and C(2, 4) in the mirror line x = 5.

  1. A is 5 − 1 = 4 squares left of the mirror line.
  2. So A' is 4 squares to the right, at 5 + 4 = 9.
  3. B is 1 square left of the line, so B' is at 5 + 1 = 6.
  4. C is 3 squares left of the line, so C' is at 5 + 3 = 8.
  5. The heights do not change. The image has corners (9, 1), (6, 1) and (8, 4).

Answer: (9, 1), (6, 1) and (8, 4)

A grid numbered 0 to 10 along the bottom and 0 to 6 on the left. A dashed mirror line stands at 5. The triangle ABC is hatched on the left. Its image A', B' and C' is dotted on the right. Dashed lines join each corner to its image across the mirror line.0123456789100123456mirror lineABCA'B'C'
A is 4 squares from the mirror line and A' is 4 squares on the other side.

See it three ways

With things

Stand a small mirror upright on a line drawn on paper. Put a pencil on one side of the line. Look in the mirror. The pencil seems to stand the same distance behind the mirror.

As a picture

A grid with a dashed mirror line. A triangle with corners P, Q and R sits on the left of the line. Its reflection with corners P', Q' and R' is the same distance on the right of the line. Dashed lines join the corners across the mirror line.mirror linePQRP'Q'R'

In symbols

A point at 1 is 4 squares from the mirror at 5. Its image is 4 squares the other side, at 5 + 4 = 9.

Try it

Make a symmetrical shape

You need: a sheet of thin paper, a pencil.

  1. Fold the paper in half and open it out. The fold is the mirror line.
  2. On one side, draw half of a shape. Let it touch the fold.
  3. Fold the paper again. Hold it up to a window.
  4. Trace the lines you can see through the paper onto the other side. Open it out.

What you will see: You see the whole shape. The two halves match, and the fold is its line of symmetry.

If you cannot try it, this is what happens.
A house shape with one dashed line down the middle, numbered 1. Both halves are the same.1

A wrong idea many people have

A diagonal of a rectangle is a line of symmetry.

Why it feels right: It cuts the rectangle into two halves of the same size and shape.

The test that breaks it: Cut out a rectangle of paper. Fold it along a diagonal. The corners do not meet, because the two halves do not lie on top of each other.

What is true: A line of symmetry gives two halves that fold onto each other exactly. Halves of equal size are not enough.

Key words

line of symmetry
a line that cuts a shape into two halves that fold exactly onto each other.
mirror line
the line a shape is reflected in.
reflection
the shape you see on the other side of a mirror line.
image
the shape that a reflection makes.

Back to your guess

The answer is A: Two ways.

You can fold a rectangle across the middle in two ways: top to bottom and left to right. A fold along a diagonal does not work. The two halves do not lie on top of each other.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. How many lines of symmetry does a square have?

    A square with no lines drawn on it.
    Tap your answer.

    1. Not this one. You found the two lines through the middle of opposite sides. There are two more, along the diagonals. Go to M3.5.6. Try another.

    2. Not this one. There is more than one fold that makes the halves match. Try the middle of the sides and the diagonals. Go to M3.5.6. Try another.

    3. Yes, that is right. A square folds onto itself across the middle two ways and along each diagonal. That is 2 + 2 = 4 lines.

    See the answer

    4

    A square folds onto itself across the middle two ways and along each diagonal. That is 2 + 2 = 4 lines.

    • If you chose A: You found the two lines through the middle of opposite sides. There are two more, along the diagonals. Go to M3.5.6.
    • If you chose B: There is more than one fold that makes the halves match. Try the middle of the sides and the diagonals. Go to M3.5.6.

  2. Which of these shapes has no line of symmetry?

    Three shapes with their names: a rectangle and a kite on top, and a parallelogram below them.rectanglekiteparallelogram
    Tap your answer.

    1. Not this one. A rectangle folds onto itself across the middle, top to bottom and left to right. Go to M3.5.6. Try another.

    2. Yes, that is right. No fold makes the two halves of a parallelogram lie on top of each other. The rectangle has 2 lines and the kite has 1.

    3. Not this one. A kite has one line of symmetry, from the top corner to the bottom corner. Go to M3.5.6. Try another.

    See the answer

    Parallelogram

    No fold makes the two halves of a parallelogram lie on top of each other. The rectangle has 2 lines and the kite has 1.

    • If you chose A: A rectangle folds onto itself across the middle, top to bottom and left to right. Go to M3.5.6.
    • If you chose C: A kite has one line of symmetry, from the top corner to the bottom corner. Go to M3.5.6.

  3. The triangle ABC is reflected in the mirror line x = 5. A is at (2, 3). Where is A'?

    A grid numbered 0 to 10 along the bottom and 0 to 7 on the left. A dashed mirror line stands at 5. The hatched triangle ABC is on the left of it. The image is not drawn.01234567891001234567mirror lineABC
    Tap your answer.

    1. Yes, that is right. A is 5 − 2 = 3 squares left of the mirror line. A' is 3 squares to the right, at 5 + 3 = 8. The height stays 3.

    2. Not this one. The mirror line is upright, so only the first number changes. The second number stays 3. Go to M3.5.6. Try another.

    3. Not this one. A is 3 squares from the mirror line, and A' is 3 squares on the other side. 5 + 3 is 8, not 7. Go to M3.5.6. Try another.

    See the answer

    (8, 3)

    A is 5 − 2 = 3 squares left of the mirror line. A' is 3 squares to the right, at 5 + 3 = 8. The height stays 3.

    • If you chose B: The mirror line is upright, so only the first number changes. The second number stays 3. Go to M3.5.6.
    • If you chose C: A is 3 squares from the mirror line, and A' is 3 squares on the other side. 5 + 3 is 8, not 7. Go to M3.5.6.

  4. How many lines of symmetry does a regular hexagon have?

    A regular hexagon.
    Tap your answer.

    1. Not this one. You found the lines through opposite corners. There are also three lines through the middles of opposite sides. Go to M3.5.6. Try another.

    2. Not this one. That is two for each side. A line of symmetry goes right across the shape, so each line is counted once. Go to M3.5.6. Try another.

    3. Yes, that is right. Three lines join opposite corners and three join the middles of opposite sides. 3 + 3 = 6.

    See the answer

    6

    Three lines join opposite corners and three join the middles of opposite sides. 3 + 3 = 6.

    • If you chose A: You found the lines through opposite corners. There are also three lines through the middles of opposite sides. Go to M3.5.6.
    • If you chose B: That is two for each side. A line of symmetry goes right across the shape, so each line is counted once. Go to M3.5.6.

  5. The triangle is reflected in the mirror line y = 3. The corner P is at (1, 4). Where is P'?

    A grid numbered 0 to 7 along the bottom and 0 to 7 on the left. A dashed mirror line lies across it at 3. A hatched triangle with corners P, Q and R is above the mirror line. The image is not drawn.0123456701234567mirror linePQR
    Tap your answer.

    1. Not this one. You moved P up by 1 again. The image is on the other side of the mirror line, below it. Go to M3.5.6. Try another.

    2. Yes, that is right. P is 4 − 3 = 1 square above the mirror line. P' is 1 square below, at 3 − 1 = 2. The first number stays 1.

    3. Not this one. That is on the mirror line itself. P is 1 square above it, so P' is 1 square below: 3 − 1 = 2. Go to M3.5.6. Try another.

    See the answer

    (1, 2)

    P is 4 − 3 = 1 square above the mirror line. P' is 1 square below, at 3 − 1 = 2. The first number stays 1.

    • If you chose A: You moved P up by 1 again. The image is on the other side of the mirror line, below it. Go to M3.5.6.
    • If you chose C: That is on the mirror line itself. P is 1 square above it, so P' is 1 square below: 3 − 1 = 2. Go to M3.5.6.

  6. A regular polygon has 7 sides. How many lines of symmetry does it have?

    See the answer

    7

    A regular polygon has one line of symmetry for each side, so 7 sides give 7 lines.

  7. From an earlier chapter

    What is each angle of a regular octagon?

    See the answer

    135°

    An octagon makes 8 − 2 = 6 triangles, so the angle sum is 6 × 180° = 1 080°. Then 1 080° ÷ 8 = 135°.

Keep it

Copy this into your exercise book.

A line of symmetry gives two halves that fold onto each other. A reflection puts every point the same distance from the mirror line on the other side. A regular polygon has as many lines of symmetry as sides.

Draw: A triangle on a grid, a dashed mirror line, and its reflection. Join each corner to its image with a dashed line.

M3.5.7 Chapter 7

Turning symmetry

A square can be turned and still look the same. How many times does it fit on itself in one full turn?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

A shape has turning symmetry when it fits on itself after a part turn. You turn it about a point in its middle. That point is the centre of turning.

Draw it first

Turn a square a quarter turn

  1. A square with a dot at its centre. Step 2: the corner A at the top right is marked, and the square is turned a quarter turn, 90°, so A moves to A' at the top left. A curved arrow shows the turn. Step 3: the turned square lies exactly on the first. The labels are A, A' and 90°.

    1 Draw the square and mark its centre with a dot.

  2. A square with a dot at its centre. Step 2: the corner A at the top right is marked, and the square is turned a quarter turn, 90°, so A moves to A' at the top left. A curved arrow shows the turn. Step 3: the turned square lies exactly on the first. The labels are A, A' and 90°.AA'90°

    2 Mark one corner A. Turn the square a quarter turn, 90°, about the centre. Corner A moves to A'. The turned square fits exactly on the first.

The order of turning symmetry is how many times a shape fits on itself in one full turn. A square fits 4 times, so its order is 4. The smallest turn that fits is 360° ÷ 4 = 90°.

A square with a dot at its centre. Four numbered dots, 1, 2, 3 and 4, mark where one corner lands after each quarter turn. The order is 4.1234
The marked corner lands on a corner 4 times, so the order is 4.

Worked example

What is the order of turning symmetry of a rectangle that is not a square?

  1. Mark one corner of the rectangle.
  2. Turn the rectangle a half turn, 180°. It fits on itself.
  3. A quarter turn does not fit. The long sides would land on the short sides.
  4. After the full turn it fits again.
  5. It fits 2 times in one full turn, so the order is 2.

Answer: 2

A rectangle with a dot at its centre. Two numbered dots, 1 and 2, mark where one corner lands after each half turn. The order is 2.12
The marked corner lands on a corner 2 times, so the order is 2.

Worked example

A regular hexagon fits on itself in a full turn. What is its order, and the smallest turn?

  1. A regular hexagon has 6 equal sides and 6 equal angles.
  2. Each corner can move to the next corner and the shape still fits.
  3. So it fits 6 times in one full turn.
  4. The order is 6.
  5. The smallest turn is 360° ÷ 6 = 60°.

Answer: Order 6 and a smallest turn of 60°

A regular hexagon with a dot at its centre. Six numbered dots, 1 to 6, mark where one corner lands after each turn of 60°. The order is 6.123456
Six turns of 60° make one full turn of 360°.

Some shapes fit only after the full turn. Their order is 1. We say they have no turning symmetry. An isosceles triangle is one. A shape can have turning symmetry and no line of symmetry.

A table with three columns: shape, lines and order. The square has 4 lines and order 4. The rectangle has 2 lines and order 2. The parallelogram has 0 lines and order 2. The kite has 1 line and order 1. The letter Z has 0 lines and order 2.shapelinesordersquare44rectangle22parallelogram02kite11letter Z02
Lines of symmetry and the order of turning symmetry are different counts.

You can use turning symmetry to make a pattern. Draw one shape. Copy it round a centre point, one turn at a time. For order 4, each copy is turned 90° from the last. You draw 4 copies in all.

Draw it first

Make a pattern of order 4

  1. A pattern built round a centre dot. Step 1: one triangle to the right of the centre. Step 2: a second triangle turned 90°. Step 3: a third triangle turned 180°. Step 4: a fourth triangle turned 270°. The finished pattern has order 4.

    1 Mark a centre point. Draw a shape beside it.

  2. A pattern built round a centre dot. Step 1: one triangle to the right of the centre. Step 2: a second triangle turned 90°. Step 3: a third triangle turned 180°. Step 4: a fourth triangle turned 270°. The finished pattern has order 4.

    2 Turn the paper 90° about the centre. Draw the shape again.

  3. A pattern built round a centre dot. Step 1: one triangle to the right of the centre. Step 2: a second triangle turned 90°. Step 3: a third triangle turned 180°. Step 4: a fourth triangle turned 270°. The finished pattern has order 4.

    3 Turn another 90°. Draw the shape a third time.

  4. A pattern built round a centre dot. Step 1: one triangle to the right of the centre. Step 2: a second triangle turned 90°. Step 3: a third triangle turned 180°. Step 4: a fourth triangle turned 270°. The finished pattern has order 4.

    4 Turn another 90°. Draw it a fourth time. The pattern has order 4.

See it three ways

With things

Trace a square on paper. Hold the tracing on the square with a pencil point at its centre. Turn the tracing a quarter turn.

As a picture

A square with a dot at its centre. The corner A moves to A' after a turn of 90°, shown by a curved arrow. The turned square lies on the first. The labels are A, A' and 90°.AA'90°

In symbols

Order 4. The smallest turn is 360° ÷ 4 = 90°.

Try it

Find the order with tracing paper

You need: a shape drawn on paper, such as a triangle with equal sides, a sheet of thin paper, a pencil.

  1. Lay the thin paper on the shape. Trace the shape and mark its centre.
  2. Hold the tracing on the shape with a pencil point on the centre.
  3. Turn the tracing slowly. Each time it fits on the shape, count 1.
  4. Stop when the tracing is back where it started. The count is the order.

What you will see: A triangle with equal sides fits 3 times in a full turn, so its order is 3. A rectangle fits 2 times.

If you cannot try it, this is what happens.
A triangle with equal sides and a dot at its centre. Three numbered dots, 1, 2 and 3, mark where one corner lands after each turn of 120°. The order is 3.123

A wrong idea many people have

A parallelogram has no symmetry.

Why it feels right: It has no line of symmetry, so it seems to have no symmetry at all.

The test that breaks it: Trace a parallelogram. Hold the tracing at the centre and turn it a half turn. It fits on the shape.

What is true: A parallelogram has turning symmetry of order 2. Line symmetry and turning symmetry are different.

Key words

turning symmetry
a shape has it when it fits on itself after a part turn.
centre of turning
the point a shape is turned about.
order
how many times a shape fits on itself in one full turn.

Back to your guess

The answer is A: Four times, every quarter turn.

Turn a square about its centre. After a quarter turn it fits on itself. It fits again after a half turn, after three quarters of a turn, and after the full turn. That is 4 times.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What is the order of turning symmetry of a triangle with three equal sides?

    A triangle with three equal sides and a dot at its centre.
    Tap your answer.

    1. Not this one. Order 1 means it fits only after the full turn. This triangle also fits after a turn of 120°. Go to M3.5.7. Try another.

    2. Not this one. There are 3 corners, and each turn moves a corner to the next corner. So there are 3 turns, not 6. Go to M3.5.7. Try another.

    3. Yes, that is right. The triangle fits on itself after 120°, 240° and 360°. That is 3 times, so the order is 3.

    See the answer

    3

    The triangle fits on itself after 120°, 240° and 360°. That is 3 times, so the order is 3.

    • If you chose A: Order 1 means it fits only after the full turn. This triangle also fits after a turn of 120°. Go to M3.5.7.
    • If you chose B: There are 3 corners, and each turn moves a corner to the next corner. So there are 3 turns, not 6. Go to M3.5.7.

  2. Which shape has turning symmetry of order 2 but no line of symmetry?

    Three shapes with their names: a rectangle and an isosceles triangle on top, and a parallelogram below them.rectangleisosceles triangleparallelogram
    Tap your answer.

    1. Not this one. A rectangle has order 2, but it also has 2 lines of symmetry. Go to M3.5.6. Try another.

    2. Yes, that is right. A parallelogram fits on itself after a half turn, so its order is 2. No fold makes its halves match, so it has no line of symmetry.

    3. Not this one. An isosceles triangle has 1 line of symmetry and its order is 1. Go to M3.5.7. Try another.

    See the answer

    Parallelogram

    A parallelogram fits on itself after a half turn, so its order is 2. No fold makes its halves match, so it has no line of symmetry.

    • If you chose A: A rectangle has order 2, but it also has 2 lines of symmetry. Go to M3.5.6.
    • If you chose C: An isosceles triangle has 1 line of symmetry and its order is 1. Go to M3.5.7.

  3. A shape fits on itself 5 times in one full turn. What is the smallest turn that fits it?

    A regular pentagon with a dot at its centre. Five numbered dots, 1 to 5, mark where one corner lands after each turn.12345
    Tap your answer.

    1. Yes, that is right. The 5 turns make one full turn of 360°. Each turn is 360° ÷ 5 = 72°.

    2. Not this one. That is the turn for order 4. Divide 360° by 5. Go to M3.5.7. Try another.

    3. Not this one. That is the turn for order 6. Divide 360° by 5. Go to M3.5.7. Try another.

    See the answer

    72°

    The 5 turns make one full turn of 360°. Each turn is 360° ÷ 5 = 72°.

    • If you chose B: That is the turn for order 4. Divide 360° by 5. Go to M3.5.7.
    • If you chose C: That is the turn for order 6. Divide 360° by 5. Go to M3.5.7.

  4. You copy a shape 4 times round a centre to make a pattern of order 4. How far do you turn it each time?

    A pattern with a dot at its centre and two triangles, the second turned from the first. The pattern is not finished.
    Tap your answer.

    1. Not this one. That would give 8 copies in a full turn. For 4 copies divide 360° by 4. Go to M3.5.7. Try another.

    2. Not this one. That is a half turn. It would give only 2 copies in a full turn. Go to M3.5.7. Try another.

    3. Yes, that is right. Four copies make one full turn of 360°. Each turn is 360° ÷ 4 = 90°.

    See the answer

    90°

    Four copies make one full turn of 360°. Each turn is 360° ÷ 4 = 90°.

    • If you chose A: That would give 8 copies in a full turn. For 4 copies divide 360° by 4. Go to M3.5.7.
    • If you chose B: That is a half turn. It would give only 2 copies in a full turn. Go to M3.5.7.

  5. Which shape has a line of symmetry but no turning symmetry?

    Three shapes with their names: a square and an isosceles triangle on top, and a parallelogram below them.squareisosceles triangleparallelogram
    Tap your answer.

    1. Not this one. A square has 4 lines of symmetry and also turning symmetry of order 4. Go to M3.5.7. Try another.

    2. Yes, that is right. An isosceles triangle has 1 line of symmetry. It fits on itself only after the full turn, so its order is 1.

    3. Not this one. A parallelogram has turning symmetry of order 2 but no line of symmetry. This question asks the other way round. Go to M3.5.7. Try another.

    See the answer

    Isosceles triangle

    An isosceles triangle has 1 line of symmetry. It fits on itself only after the full turn, so its order is 1.

    • If you chose A: A square has 4 lines of symmetry and also turning symmetry of order 4. Go to M3.5.7.
    • If you chose C: A parallelogram has turning symmetry of order 2 but no line of symmetry. This question asks the other way round. Go to M3.5.7.

  6. A regular polygon has 6 sides. What is its order of turning symmetry?

    See the answer

    6

    A regular polygon fits on itself once for each of its corners in a full turn. 6 sides give order 6.

  7. From an earlier chapter

    The point (2, 3) is reflected in the mirror line x = 6. Where is the image?

    See the answer

    (10, 3)

    The point is 6 − 2 = 4 squares left of the mirror line. The image is 4 squares right, at 6 + 4 = 10. The height stays 3.

Keep it

Copy this into your exercise book.

The order of turning symmetry is how many times a shape fits on itself in one full turn. The smallest turn is 360° ÷ order. A pattern of order 4 copies a shape every 90°.

Draw: A square with a dot at its centre. Mark one corner 1, 2, 3 and 4 as it turns a quarter turn each time.

M3.5.8 Chapter 8

Solids, nets and views

A cube has 6 faces, 8 corners and 12 edges. Is there a rule that links the three numbers?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

A solid has flat faces. Two faces meet along an edge. Edges meet at a corner. A cube has 6 faces, 12 edges and 8 corners.

Isometric dotted paper has dots in a pattern of triangles. You draw lines that go straight up, up to the right and up to the left. Each step from dot to dot is 1 cm here.

Draw it first

Draw a cube on dotted paper

  1. A cube drawn on isometric dotted paper, step by step. Each edge is 3 cm. Step 1: one upright line. Step 2: two lines up to the right and up to the left from its foot, and an upright line at the end of each. Step 3: the top, made with four more lines. Step 4: the finished cube shows 9 lines, each 3 cm long. The labels are 3 cm, 3 cm and 3 cm.

    1 Draw one upright line, 3 steps long.

  2. A cube drawn on isometric dotted paper, step by step. Each edge is 3 cm. Step 1: one upright line. Step 2: two lines up to the right and up to the left from its foot, and an upright line at the end of each. Step 3: the top, made with four more lines. Step 4: the finished cube shows 9 lines, each 3 cm long. The labels are 3 cm, 3 cm and 3 cm.

    2 From its foot draw a line up to the right and a line up to the left. Each is 3 steps long. Add an upright line at the end of each.

  3. A cube drawn on isometric dotted paper, step by step. Each edge is 3 cm. Step 1: one upright line. Step 2: two lines up to the right and up to the left from its foot, and an upright line at the end of each. Step 3: the top, made with four more lines. Step 4: the finished cube shows 9 lines, each 3 cm long. The labels are 3 cm, 3 cm and 3 cm.

    3 Join the tops to make the top face. Use four lines in all, each 3 steps long.

  4. A cube drawn on isometric dotted paper, step by step. Each edge is 3 cm. Step 1: one upright line. Step 2: two lines up to the right and up to the left from its foot, and an upright line at the end of each. Step 3: the top, made with four more lines. Step 4: the finished cube shows 9 lines, each 3 cm long. The labels are 3 cm, 3 cm and 3 cm.3 cm3 cm3 cm

    4 Check: 9 lines, each 3 steps long. The corner at the back is hidden right behind the front top corner.

You can draw a cuboid in the same way. Its three lengths can be different. A cuboid is a box.

A cuboid on isometric dotted paper. The top face is shaded with dots and is labelled top. The face on the right is labelled front. The face on the left is labelled side and is shaded with hatching. The length is 4 cm, the width is 2 cm and the height is 3 cm.topsidefront4 cm2 cm3 cm
A cuboid 4 cm long, 2 cm wide and 3 cm high.

Now count the parts of a cube. Letters name the corners. Dashed lines show the edges you cannot see.

A cube turned a little so all 8 corners can be seen. Each corner is marked with a dot and named A, B, C, D, E, F, G and H. Nine edges are solid and three edges at the back are dashed.ABDCEFHG
A cube has 8 corners, 12 edges (9 solid and 3 dashed) and 6 faces. This cube is turned a little and is not on dotted paper, so the hidden corner shows.

Take a solid with flat faces and no hole through it. Count its faces, corners and edges. Then work out faces + corners − edges. The answer is always 2.

Worked example

Check that faces + corners − edges = 2 for a cube.

  1. A cube has 6 faces.
  2. It has 8 corners.
  3. It has 9 edges you can see and 3 at the back, so 12 edges.
  4. Faces + corners − edges = 6 + 8 − 12 = 2.

Answer: 2

A square based pyramid with 5 corners named A, B, C, D and E. The base has corners A, B, C and D. The top corner is E. The edge at the back is dashed.ABDCE
A square based pyramid: 1 square face and 4 triangle faces.

Worked example

Check that faces + corners − edges = 2 for a square based pyramid.

  1. It has 1 square face and 4 triangle faces. That is 5 faces.
  2. It has 4 corners on the base and 1 at the top. That is 5 corners.
  3. It has 4 edges on the base and 4 edges up to the top. That is 8 edges.
  4. Faces + corners − edges = 5 + 5 − 8 = 2.

Answer: 2

A triangular prism on isometric dotted paper with 6 corners named A, B, C, D, E and F. The triangle at the front has corners A, B and C. The triangle at the back has corners D, E and F. The edges at the back are dashed.ABCDEF
A triangular prism: 2 triangle faces and 3 rectangle faces.

Worked example

Check that faces + corners − edges = 2 for a triangular prism.

  1. It has 2 triangle faces and 3 rectangle faces. That is 5 faces.
  2. It has 3 corners at each end. That is 6 corners.
  3. It has 3 edges at each end and 3 edges between the ends. That is 9 edges.
  4. Faces + corners − edges = 5 + 6 − 9 = 2.

Answer: 2

A solid looks different from the front, from the side and from above. Each of these is a view. You draw a view as a flat shape made of squares.

A stack of 9 cubes drawn on isometric dotted paper. Three towers stand in a row at the front, with 3, 1 and 2 cubes. Two more towers, with 2 cubes and 1 cube, stand behind the tower of 3.
A stack of 9 cubes on dotted paper.

Worked example

How many squares are in the front view of the stack in the picture?

  1. The front view has one column for each column of the stack. There are 3 columns.
  2. Stand at the front. The left column has a tallest tower of 3 cubes.
  3. The middle column has a tallest tower of 1 cube.
  4. The right column has a tallest tower of 2 cubes.
  5. The front view has 3 + 1 + 2 = 6 squares.

Answer: 6

The front, side and top views of the stack, drawn on grids of squares. The front view has 6 squares, the side view has 6 squares and the top view has 5 squares. In the top view the squares show the numbers 1, 2, 3, 1 and 2.frontsidetop12312
Each view is a flat shape. The top view shows how many cubes stand on each square.

A net is a flat shape that folds up into a solid. The net of a cube is 6 squares joined along their sides. Not every set of 6 squares folds into a cube.

The net of a cuboid 4 cm long, 3 cm wide and 2 cm high. It has 6 rectangles numbered 1 to 6. Faces 1, 2, 3 and 4 are in a row. Face 5 is above face 2 and face 6 is below it. The edges are labelled 4 cm, 3 cm and 2 cm. Dashed lines are the folds.1234564 cm3 cm2 cm
The net of a cuboid. Dashed lines are folds. Opposite faces have the same shading.

See it three ways

With things

Take a small box, such as a tea box. Count its flat faces, then its edges, then its corners.

As a picture

A cuboid with its 8 corners named A, B, C, D, E, F, G and H. Three edges at the back are dashed.ABDCEFHG

In symbols

Faces + corners − edges = 6 + 8 − 12 = 2.

Try it

Count the parts of a box

You need: a small box, such as a tea box or a matchbox, a pencil, a sheet of paper.

  1. Count the flat faces of the box. Write the number.
  2. Count the edges. An edge is a line where two faces meet.
  3. Count the corners.
  4. Work out faces + corners − edges.

What you will see: A box has 6 faces, 8 corners and 12 edges. The answer to the sum is 2.

If you cannot try it, this is what happens.
A cuboid on isometric dotted paper. The top face is labelled top, the face on the right is labelled front and the face on the left is labelled side.topsidefront

A wrong idea many people have

To count the edges of a solid, count the lines you can see in the picture.

Why it feels right: A picture shows lines, so counting them seems enough.

The test that breaks it: Count the solid lines on the picture of the cube. You find 9. Now count the dashed lines too. There are 3 more, so the cube has 12 edges.

What is true: Edges at the back are drawn as dashed lines. Count the dashed lines too.

Key words

face
a flat surface of a solid.
edge
a line where two faces of a solid meet.
corner
a point where edges of a solid meet.
net
a flat shape that folds up into a solid.
view
the flat shape you see from the front, the side or the top.

Back to your guess

The answer is C: Yes. Faces plus corners take away edges is 2 for every solid like this.

For a cube, 6 + 8 − 12 = 2. The same sum gives 2 for a pyramid and for a prism. It holds for solids with flat faces and no hole through them. Adding the three numbers gives 26 for a cube, but it does not give 26 for other solids.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A cube has edges of 4 cm. Each step from dot to dot is 1 cm. How many steps long is each line on the dotted paper?

    The first upright line of a cube drawn on isometric dotted paper. It runs from one dot up to a dot 4 steps higher, so it passes over 5 dots in all.
    Tap your answer.

    1. Yes, that is right. Each step is 1 cm, so an edge of 4 cm is 4 ÷ 1 = 4 steps long. It passes over 5 dots, but the steps are what count.

    2. Not this one. You counted the dots. A line that is 4 cm long goes over 5 dots but is only 4 steps long. Go to M3.5.8. Try another.

    3. Not this one. That is the size of the cube drawn in the chapter. This cube has edges of 4 cm. Go to M3.5.8. Try another.

    See the answer

    4

    Each step is 1 cm, so an edge of 4 cm is 4 ÷ 1 = 4 steps long. It passes over 5 dots, but the steps are what count.

    • If you chose B: You counted the dots. A line that is 4 cm long goes over 5 dots but is only 4 steps long. Go to M3.5.8.
    • If you chose C: That is the size of the cube drawn in the chapter. This cube has edges of 4 cm. Go to M3.5.8.

  2. How many edges has a square based pyramid?

    A square based pyramid with 5 corners named A, B, C, D and E. The edge at the back is dashed.ABDCE
    Tap your answer.

    1. Not this one. That is the number of faces, and also the number of corners. Count the lines: 4 on the base and 4 up to the top. Go to M3.5.8. Try another.

    2. Yes, that is right. The base has 4 edges and 4 more edges go up to the top corner. 4 + 4 = 8.

    3. Not this one. That is the edges of the base only. Add the 4 edges that go up to the top corner. Go to M3.5.8. Try another.

    See the answer

    8

    The base has 4 edges and 4 more edges go up to the top corner. 4 + 4 = 8.

    • If you chose A: That is the number of faces, and also the number of corners. Count the lines: 4 on the base and 4 up to the top. Go to M3.5.8.
    • If you chose C: That is the edges of the base only. Add the 4 edges that go up to the top corner. Go to M3.5.8.

  3. A triangular prism has 5 faces and 6 corners. How many edges has it? Use faces + corners − edges = 2.

    Tap your answer.

    1. Not this one. You added the 2. The rule says faces + corners − edges = 2, so edges = faces + corners − 2. Go to M3.5.8. Try another.

    2. Not this one. That is 5 + 6. Take away the 2 as well. Go to M3.5.8. Try another.

    3. Yes, that is right. Faces + corners − edges = 2, so edges = faces + corners − 2. 5 + 6 − 2 = 9.

    See the answer

    9

    Faces + corners − edges = 2, so edges = faces + corners − 2. 5 + 6 − 2 = 9.

    • If you chose A: You added the 2. The rule says faces + corners − edges = 2, so edges = faces + corners − 2. Go to M3.5.8.
    • If you chose B: That is 5 + 6. Take away the 2 as well. Go to M3.5.8.

  4. Which of these is a net of a cube?

    Three flat shapes made of 6 squares, named A, B and C. Net A is a block of 2 rows of 3 squares. Net B is a straight row of 6 squares. Net C is a cross: a row of 4 squares with one square above the second and one below it.ABC
    Tap your answer.

    1. Yes, that is right. Fold the four squares in the row round the cube. The square above and the square below fold on to the top and the bottom. Every face is covered once.

    2. Not this one. A straight row of 6 squares wraps round the cube. The fifth square lands on the first, and the sixth on the second. Go to M3.5.8. Try another.

    3. Not this one. A block of 2 rows of 3 squares will not fold. Two of the squares would land on the same face of the cube. Go to M3.5.8. Try another.

    See the answer

    Net C

    Fold the four squares in the row round the cube. The square above and the square below fold on to the top and the bottom. Every face is covered once.

    • If you chose B: A straight row of 6 squares wraps round the cube. The fifth square lands on the first, and the sixth on the second. Go to M3.5.8.
    • If you chose C: A block of 2 rows of 3 squares will not fold. Two of the squares would land on the same face of the cube. Go to M3.5.8.

  5. Look at the stack of cubes. The white faces on the right are the front. How many squares are in the front view?

    A stack of 4 cubes on isometric dotted paper. At the front there is a tower of 2 cubes with a tower of 1 cube to its right. One cube stands behind the tower of 2.
    Tap your answer.

    1. Not this one. That counts the cube behind the tower of 2 as a third column, or counts every cube. The front is the white faces on the right. The cube behind adds no square. Go to M3.5.8. Try another.

    2. Yes, that is right. The left column of the front view is as tall as its tallest tower, 2 squares. The right column has 1 square. 2 + 1 = 3 squares.

    3. Not this one. That is the height of the tallest tower. The front view also has a column for the tower of 1 cube. Go to M3.5.8. Try another.

    See the answer

    3

    The left column of the front view is as tall as its tallest tower, 2 squares. The right column has 1 square. 2 + 1 = 3 squares.

    • If you chose A: That counts the cube behind the tower of 2 as a third column, or counts every cube. The front is the white faces on the right. The cube behind adds no square. Go to M3.5.8.
    • If you chose C: That is the height of the tallest tower. The front view also has a column for the tower of 1 cube. Go to M3.5.8.

  6. A solid has 7 faces and 15 edges. How many corners has it? Use faces + corners − edges = 2.

    Tap your answer.

    1. Not this one. You added 7 and 15 and took away 2. Put the rule in order: corners = 2 + edges − faces. Go to M3.5.8. Try another.

    2. Not this one. You took 2 away instead of adding it. Corners = 2 + edges − faces. Go to M3.5.8. Try another.

    3. Yes, that is right. Faces + corners − edges = 2, so corners = 2 + edges − faces. 2 + 15 − 7 = 10.

    See the answer

    10

    Faces + corners − edges = 2, so corners = 2 + edges − faces. 2 + 15 − 7 = 10.

    • If you chose A: You added 7 and 15 and took away 2. Put the rule in order: corners = 2 + edges − faces. Go to M3.5.8.
    • If you chose B: You took 2 away instead of adding it. Corners = 2 + edges − faces. Go to M3.5.8.

  7. A solid has 6 faces and 12 edges. How many corners has it?

    See the answer

    8

    Corners = 2 + edges − faces = 2 + 12 − 6 = 8.

  8. From an earlier chapter

    A shape has turning symmetry of order 8. What is the smallest turn that fits it?

    See the answer

    45°

    The 8 turns make one full turn of 360°. Each turn is 360° ÷ 8 = 45°.

Keep it

Copy this into your exercise book.

A solid has faces, edges and corners. Faces + corners − edges = 2. A net folds into a solid. A view is the flat shape you see from the front, side or top.

Draw: A cube on dotted paper, with the 9 edges you can see. Write 6 + 8 − 12 = 2 beside it.

Key words, A to Z

Every key word in this book, what it means, and the chapter that teaches it.

alternate angles
angles between the lines on opposite sides of the transversal, in a Z shape. Between parallel lines they are equal. M3.5.2
angle sum
the total of the angles in a shape. For a triangle it is 180°. M3.5.3
angles on a line
angles that fill a straight line. They add to 180°. M3.5.1
angles round a point
angles that fill a full turn. They add to 360°. M3.5.1
arc
a piece of a circle. M3.5.5
bisect
to cut exactly in half. The line that bisects an angle is the angle bisector. M3.5.5
centre of turning
the point a shape is turned about. M3.5.7
compasses
a tool that draws circles and arcs. It has a sharp point. M3.5.5
construction
a drawing made with a ruler and compasses, with the arcs left in. M3.5.5
corner
a point where edges of a solid meet. M3.5.8
corresponding angles
angles in the same place at each crossing, in an F shape. Between parallel lines they are equal. M3.5.2
diagonal
a line that joins two corners that are not next to each other. M3.5.4
edge
a line where two faces of a solid meet. M3.5.8
equilateral triangle
a triangle with three equal sides and three angles of 60°. M3.5.3
face
a flat surface of a solid. M3.5.8
image
the shape that a reflection makes. M3.5.6
isosceles triangle
a triangle with two equal sides and two equal angles. M3.5.3
line of symmetry
a line that cuts a shape into two halves that fold exactly onto each other. M3.5.6
mirror line
the line a shape is reflected in. M3.5.6
net
a flat shape that folds up into a solid. M3.5.8
order
how many times a shape fits on itself in one full turn. M3.5.7
parallel lines
straight lines that stay the same distance apart and never meet. M3.5.2
polygon
a flat shape with straight sides. With n sides, its angles add to (n − 2) × 180°. M3.5.4
quadrilateral
a shape with four straight sides. Its angles add to 360°. M3.5.4
reflection
the shape you see on the other side of a mirror line. M3.5.6
regular polygon
a polygon with all sides equal and all angles equal. M3.5.4
right angled triangle
a triangle with one angle of 90°. M3.5.3
scalene triangle
a triangle with no equal sides. M3.5.3
transversal
a straight line that crosses two or more other lines. M3.5.2
turning symmetry
a shape has it when it fits on itself after a part turn. M3.5.7
vertex
the point where two lines meet. M3.5.1
vertically opposite angles
angles that face each other where two straight lines cross. They are equal. We also call them opposite angles. M3.5.1
view
the flat shape you see from the front, the side or the top. M3.5.8