Year 7 · Unit 4 Angles and constructions · Lesson 1 of 6

How do we name an angle and say how big it is?

Two arms and a vertex make an angle. Open one, name it, sort it by size, then read its size from a protractor scale without being fooled by the second scale.

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 1, lesson 1 (place value and decimals): which is greater, 0.4 or 0.35?
    Answer0.4. It has 4 tenths and 0.35 has 3, so the tenths decide.
  2. From Unit 1, lesson 6 (order of operations): what is 180 − 2 × 35?
    Answer110. Multiply first: 2 × 35 = 70, then 180 − 70 = 110.
  3. From Unit 1, lesson 7 (rounding and estimating): round 148 to the nearest 10.
    Answer150. The ones digit is 8, so the tens round up.
Words used in this lesson
vertex
the point where two arms of an angle meet. For example: In ∠ABC the vertex is B.
ray
a straight line that starts at a point and goes on for ever one way. For example: Ray AB starts at A and passes through B.
acute, obtuse
less than 90°; between 90° and 180°. For example: 30° is acute; 150° is obtuse.
right angle
an angle of exactly 90°, a square corner. For example: The corner of a page.
reflex
an angle bigger than 180° and smaller than 360°. For example: The big angle the other way round from 40° is 320°.
protractor
a tool with scales for measuring and drawing angles. For example: A half circle marked 0 to 180.

Two arms and a vertex

A kite-maker in Gujarat bends two bamboo strips and ties them where they cross. How wide the strips open decides how the kite flies, and the maker needs a way to say “this wide” exactly. Geometry starts with a few agreed words and symbols:

Predict first: as the arm turns from 0° to 360°, at which sizes does the name of the angle change? Write down your guesses, then slide.

40°
40°size of ∠ABC
acutetype of angle
320°the other way round

The angle opens 40°. That is less than 90°, so it is acute. Measured the other way round the same two arms make 320°.

Show me the steps
  1. ∠ABC = 40°. It is less than 90°, so it is acute. The other way round it is 360 − 40 = 320°.
  2. Open it to 90°. The arms are at a square corner, which is a right angle. The other way round is 270°.
  3. Open it to 140°. It is more than 90° and less than 180°, so it is obtuse. The other way round is 220°.
  4. Open it to 180°. The arms point in opposite directions and make a straight line. The other way round is also 180°.
  5. Open it to 250°. It is more than 180° and less than 360°, so it is a reflex angle. The other way round is 110°.

An angle measures how far one arm has turned from the other, not how long the arms are. Angles are measured in degrees: a full turn is 360° and a half turn is 180°. Every angle has a name for its size.

  • Acute: more than 0° and less than 90°.
  • Right: exactly 90°, a square corner.
  • Obtuse: more than 90° and less than 180°.
  • Straight: exactly 180°.
  • Reflex: more than 180° and less than 360°.

Two arms make two angles, one each way round, and together they make a full turn of 360°. Back to the kite: strips opened 70° make an acute angle, and the other way round they make 360 − 70 = 290°.

Reading a protractor

A protractor has two scales that run in opposite directions, so every arm crosses two numbers. The rule is short: put the centre on the vertex, lay one arm on a 0 line, and read the scale that starts at 0 on that arm. Then check the number against the picture. If the angle looks smaller than a square corner, the answer must be below 90. Choose which way arm BA points and watch both scales.

65°
outer scalescale with 0 on arm BA
65°size of ∠ABC
115°the other scale reads

Arm BA lies on the 0 of the outer scale, so the outer scale gives the size: 65°. The inner scale reads 115°, which is 180° − 65°, and is the wrong scale for this angle.

Show me the steps
  1. Arm BA points right and ∠ABC = 65°. The outer scale starts at 0 on BA and reads 65°. The inner scale reads 180 − 65 = 115°.
  2. Press the button so BA points left. It is the same 65° angle. Now the inner scale has its 0 on BA and reads 65°, while the outer scale reads 115°.
  3. Set the size to 140° with BA pointing right. The outer scale reads 140° and the inner scale reads 40°. The angle is obtuse, so 140° is sensible and 40° is not.
  4. Set the size to 90°. Both scales read 90°, the one angle where the choice of scale cannot matter.

A common mistake: “The arm crosses 115 on one scale and 65 on the other, so I will take the one I read first.” Two readings that add up to 180 are the signal to stop and decide which scale has its 0 on your arm. Estimate acute or obtuse first, and the wrong reading will not fit.

Drawing an angle

Drawing runs the same routine backwards. To draw ∠ABC = 47°: draw arm BA, put the centre of the protractor on B with its 0 on BA, make a mark at 47 on the scale that starts at 0, then join B to the mark. Your turn: draw ∠ABC = 47° and ∠PQR = 132°, swap with a partner and measure each other's angles to the nearest degree.

Worked example: name, read and classify

Arm QP lies on the 0 of the inner scale. Arm QR crosses the inner scale at 128. Name the angle, give its size and classify it.

  1. The vertex is Q, the middle letter, so the angle is ∠PQR.
  2. The 0 on arm QP is on the inner scale, so the inner scale gives the size: 128°.
  3. Check with the other scale: it would read 180 − 128 = 52. The picture opens wider than a square corner, so 128 fits and 52 does not.
  4. 128 is between 90 and 180, so ∠PQR is obtuse, 128°.

Your go. The first two steps are given. Arm XY lies on the 0 of the outer scale and arm XZ passes through 73 on the outer scale. Name the angle, classify it, and give the reflex angle the arms also make.

  1. The vertex is X, the middle letter, so the angle is ∠YXZ.
  2. The 0 on arm XY is on the outer scale, so the outer scale gives the size: 73°.
  3. Now classify 73° and find the angle the other way round yourself, then check.
Check the full answer

73° is less than 90°, so ∠YXZ is acute. The two arms make a full turn of 360°, so the reflex angle is 360 − 73 = 287°.

Check before you go on

1. An angle measures 215°. What type of angle is it?

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. Arm QP lies on the 0 of the inner scale of a protractor. Arm QR crosses the outer scale at 35. What is the size of ∠PQR in degrees?

3. Which of these angles is obtuse?

Level 2 · Building

4. A minute hand starts at 12 and turns clockwise to 9, which is 45 minutes. Through how many degrees has it turned?

5. How many degrees more than a right angle is an angle of 133°?

6. Which of these says “line AB is perpendicular to line MN”?

Level 3 · Stretch

7. At 3:30 the minute hand points at 6 and the hour hand is halfway between 3 and 4. What is the smaller angle between the two hands, in degrees? (Each hour mark is 30° from the next.)

8. Two arms make a reflex angle that is exactly 4 times the size of the other angle they make. How big is the reflex angle?

Something to think about

Some surveyors split a full turn into 400 equal units instead of 360. In that system, how many units is a right angle, and how many is a straight angle? Which of 360 and 400 can be shared into more equal whole-number parts, and why might that matter to a designer?

One way to think about it

A right angle is a quarter of 400, so 100 units, and a straight angle is 200 units. 360 has 24 factors (it is 2³ × 3² × 5) and 400 has 15 (24 × 5²), so 360 splits into more equal whole-number angles. That is the same idea as the factors and prime factorisation of Unit 1.

Exit ticket

9. Sofia reads 62° on a protractor for an angle that is clearly obtuse. What has she most likely done?