Chapter 1 of 8

All 8 chapters
  1. 1Angles on a line and round a point, readunderstood
  2. 2Angles and parallel lines, readunderstood
  3. 3Triangles, readunderstood
  4. 4Angles in polygons, readunderstood
  5. 5Drawing with ruler and compasses, readunderstood
  6. 6Line symmetry and reflection, readunderstood
  7. 7Turning symmetry, readunderstood
  8. 8Solids, nets and views, readunderstood

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

  • Find a missing angle on a straight line
  • Find a missing angle round a point
  • Find the opposite angles where two straight lines cross

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.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (801 KB). It opens on any phone.