Year 7 · Unit 4 Angles and constructions · Lesson 2 of 6

How can we find an angle without measuring it?

Angles on a straight line add up to 180°, angles at a point add up to 360°, and where two lines cross the opposite angles are equal. Turn each fact into a reason you can write down.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Unit 4, lesson 1 (points, lines and angles): what type of angle is 135°?
    AnswerObtuse. It is more than 90° and less than 180°.
  2. From Unit 1, lesson 6 (order of operations): what is 360 − (90 + 2 × 45)?
    Answer180. Brackets first: multiply 2 × 45 = 90, add 90 to make 180, then 360 − 180 = 180.
  3. From Unit 1, lesson 3 (factors, multiples and primes): is 180 a multiple of 9?
    AnswerYes. 9 × 20 = 180, and the digits 1 + 8 + 0 add to 9.
Words used in this lesson
straight line
a line with no bend; as an angle it is exactly 180°. For example: A ruler's edge.
vertically opposite
the pair of angles across the crossing point from each other. For example: a and c in the first figure.
supplementary
two angles that add up to 180°. For example: 47° and 133°.
complementary
two angles that add up to 90°. For example: 38° and 52°.
adjacent
next to each other, sharing an arm and a vertex. For example: a and b in the first figure.
reason
the fact that makes a step true. For example: angles on a straight line add up to 180°.

Angles that have to fit

In Marrakech a tile-setter lays eight pointed tiles round the centre of a star. They must close with no gap and no overlap, so the angles at the centre have to add up to a whole turn. That is a fact about every point, and it lets you find an angle without a protractor. Cross two lines and watch the four angles.

Predict first: a is 65°. What is the angle next to it on the line? What is the angle opposite it? Then slide a and see if your rule still holds when a is 30° and when it is 150°.

65°
65°angle a (and c)
115°angle b (and d)
180°a + b

a = 65° and the angle opposite it, c, is also 65°. The angle next to a on the line is b = 180° − 65° = 115°, and d = 115° as well.

Show me the steps
  1. a = 65°. The angle next to it on the straight line is b = 180 − 65 = 115°. The opposite angles are equal, so c = 65° and d = 115°.
  2. Move a to 90°. Then b = 90° too, and all four angles are right angles: the lines are perpendicular.
  3. Move a to 30°. The opposite angle c is 30°, and b and d are both 150°. a + b = 180° still.
  4. Move a to 150°. Now a and c are the large pair, 150°, and b and d are 30°. a + b = 180° still.

Three facts, each one a reason you can write.

  • Angles on a straight line add up to 180°. A straight line is a half turn.
  • Angles at a point add up to 360°. They go once all the way round.
  • Vertically opposite angles are equal. They are the pair opposite each other where two lines cross.

Why opposite angles are equal: a + b = 180 and c + b = 180 (both pairs are on a straight line), so a = c.

Two angles that add up to 180° are supplementary; two that add up to 90° are complementary. Each fact is a total, so a missing angle is the total minus the angles you know. Write the reason beside every step.

Filling the missing angle

Choose half a turn or a full turn, set the two angles you know, and find the last one by subtraction. The total is the only thing that changes between a line and a point.

60°
50°
180°the total
110°p + q
70°the missing angle x

On a straight line the total is 180°. p + q = 60° + 50° = 110°, so the missing angle is x = 180° − 110° = 70°.

Show me the steps
  1. On a straight line with p = 60° and q = 50°: p + q = 110°, so x = 180 − 110 = 70°.
  2. Press Round a point with the same p and q. The total is now 360°, so x = 360 − 110 = 250°, a reflex angle.
  3. On a line with p = 100° and q = 70°: p + q = 170°, so x = 180 − 170 = 10°, a very thin angle.
  4. Round a point with p = 100° and q = 70°: x = 360 − 170 = 190°.

Back to the star: eight equal tiles share the full turn of 360°, so each one meets the centre at 360 ÷ 8 = 45°.

A common mistake: “The angle next to 65° on a line is 25°, because 90 − 65 = 25.” 90 is the total for a right angle, not a line. On a straight line the total is 180, so the next angle is 180 − 65 = 115. Ask: is this a line, a point or a right angle? Then pick the total.

Also, a diagram in a question is often marked “not drawn to scale”. Find the angles from the facts, never by measuring the picture. A protractor is for the angles you draw yourself.

Worked example: an unknown in the angles

Angles on a straight line are 2x°, (3x + 10)° and 70°. Find x, then find each angle.

  1. They are on a straight line, so they add up to 180°: 2x + (3x + 10) + 70 = 180.
  2. Collect like terms: 5x + 80 = 180.
  3. Subtract 80 from both sides: 5x = 100, so x = 20.
  4. The angles are 2 × 20 = 40°, 3 × 20 + 10 = 70° and 70°. Check: 40 + 70 + 70 = 180. x = 20

Your go. The first two steps are given. Angles round a point are 3x°, (x + 20)°, 110° and 90°. Find x and the largest angle.

  1. They are at a point, so they add up to 360°: 3x + (x + 20) + 110 + 90 = 360.
  2. Collect like terms: 4x + 220 = 360.
  3. Now solve for x yourself and find each angle. Then check that they add up to 360.
Check the full answer

4x = 140, so x = 35. The angles are 3 × 35 = 105°, 35 + 20 = 55°, 110° and 90°. The largest is 110°, and 105 + 55 + 110 + 90 = 360.

Check before you go on

1. Two straight lines cross. One of the four angles is 65°. What are the other three?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Two angles are next to each other on a straight line. One is 118°. What is the other, in degrees?

3. Three angles meet at a point and fill the whole turn. Two of them are 140° and 95°. What is the third, in degrees?

Level 2 · Building

4. Two angles are complementary. One is 38°. What is the other, in degrees?

5. Two straight lines cross. Two vertically opposite angles are (4x − 5)° and 75°. Find x.

6. Two angles measure 47° and 133°. Which word describes the pair?

Level 3 · Stretch

7. Three angles are on a straight line: (2x + 15)°, (x + 30)° and (3x − 15)°. Find x. (2x + 15)°(x + 30)°(3x − 15)°not drawn to scale

8. Two angles are next to each other on a straight line. One is 3 times as big as the other. How big is the larger angle, in degrees?

Something to think about

Three straight lines cross at one point and make six angles round it. Three angles next to each other measure 40°, 55° and x. Find x, then write down all six angles. What do the six add up to, and why?

One way to think about it

The three neighbours fill half a turn, so x = 180 − (40 + 55) = 85°. The opposite angles are equal in pairs, so the other three are 40°, 55° and 85°. All six add up to 40 + 55 + 85 + 40 + 55 + 85 = 360°, which is a full turn: the six angles go once round the point.

Exit ticket

9. Kofi says: “When two lines cross, the angles opposite each other add up to 180°.” Which reply is right?