Key ideas
| Type | Size |
|---|---|
| Acute | less than 90° |
| Right angle | exactly 90° |
| Obtuse | between 90° and 180° |
| Straight | exactly 180° |
| Reflex | between 180° and 360° |
- Name an angle with three letters, like ∠ABC. The middle letter is the corner (vertex).
- Complementary angles add up to 90°. Supplementary angles (on a straight line) add up to 180°.
- Where two straight lines cross, vertically opposite angles (an X shape) are equal.
- The three angles in a triangle add up to 180°.
- When a line crosses two parallel lines: corresponding angles (an F shape) are equal, and alternate angles (a Z shape) are equal.
Types of angles
What type of angle is this? It measures 130°.(1 mark)
Compare it with a right angle.130° is bigger than 90°
In plain EnglishA right angle is a square corner (90°). This angle opens wider than a square corner.
Compare it with a straight line.130° is less than 180°
In plain EnglishIt doesn't open out as far as a straight line (180°).
Name it.Between 90° and 180°, so it is obtuse✓1
In plain EnglishAny angle between 90° and 180° is obtuse.
Naming angles
Use three letters to name the marked angle.(1 mark)
Find the vertex.The angle is at the corner B
In plain EnglishThe vertex is the point where the two arms meet. The marked angle is at B.
Read along the arms.∠ABC (or ∠CBA)✓1
In plain EnglishStart at a letter on one arm (A), go to the vertex (B), then finish on the other arm (C). B must be in the middle.
⚠ Common mistake. Writing ∠BAC or ∠ACB. The vertex letter always goes in the middle.
Complementary angles
Find the size of angle x.(2 marks)
Spot the right angle.The two angles make a right angle: 90°
In plain EnglishThe small square in the corner means 90°. So the two angles are complementary: they add to 90°.
Subtract.x = 90° − 35°✓1x = 55°✓2
In plain English90 − 35 = 55. Check: 35 + 55 = 90 ✓.
Supplementary angles
Find the size of angle x.(2 marks)
Spot the straight line.Angles on a straight line add to 180°
In plain EnglishThe two angles sit on a straight line, so they are supplementary.
Subtract.x = 180° − 128°✓1x = 52°✓2
In plain English180 − 128 = 52. Check: 128 + 52 = 180 ✓.
Vertically opposite angles
Two straight lines cross. Find the size of angle x.(1 mark)
Look for an X shape.50° and x are opposite each other where the lines cross
In plain EnglishWhen two straight lines cross they make an X. The angles facing each other across the middle are vertically opposite.
Vertically opposite angles are equal.x = 50° (vertically opposite angles)✓1
In plain EnglishSo x is the same as the angle opposite it: 50°. The other two angles are both 180° − 50° = 130°.
Give a reason: write “vertically opposite angles”.
Angles in a triangle
Find the size of angle x.(2 marks)
Add the angles you know.65° + 48° = 113°
In plain EnglishTwo of the three angles are given. Add them together.
Take away from 180°.x = 180° − 113°✓1x = 67°✓2
In plain EnglishThe three angles in a triangle always add to 180°. Check: 65 + 48 + 67 = 180 ✓.
Corresponding angles
The two horizontal lines are parallel (shown by the arrows). Find angle x.(1 mark)
Look for an F shape.70° and x are in the same position at each crossing
In plain EnglishBoth angles are above the line and to the right of the slanting line. Trace an F shape through them.
Corresponding angles are equal.x = 70° (corresponding angles)✓1
In plain EnglishBecause the lines are parallel, the slanting line crosses each one at the same angle.
Give a reason: write “corresponding angles” after your answer.
Alternate angles
The two horizontal lines are parallel. Find angle x.(1 mark)
Look for a Z shape.120° and x are on opposite sides, between the parallel lines
In plain EnglishBoth angles are inside the parallel lines, on opposite sides of the slanting line. Trace a Z shape through them.
Alternate angles are equal.x = 120° (alternate angles)✓1
In plain EnglishAlternate angles are equal when the lines are parallel.
Give a reason: write “alternate angles”.
Finding several angles
Two straight lines cross. Find the sizes of angles a, b and c.(3 marks)
Start with an angle on a straight line.a = 180° − 55° = 125°✓1(angles on a straight line)
In plain Englisha and 55° sit together on the horizontal straight line, so they add to 180°.
Use vertically opposite angles.b = 55°✓2(vertically opposite 55°)
In plain Englishb is opposite the 55° angle across the X, so it is equal.
One more vertically opposite pair.c = 125°✓3(vertically opposite a)
In plain Englishc is opposite a, so it is also 125°. Check: 125 + 55 + 55 + 125 = 360° ✓ (all the way round a point).
Tip. Work out the easiest angle first, then use it to find the next one. Write the reason beside each angle.
Estimating angles
Estimate the size of this angle.(1 mark)
Compare with angles you know.It's less than a right angle (90°)It's a bit less than half a right angle (45°)
In plain EnglishPicture a square corner. This angle is acute, and a bit narrower than halfway to a right angle.
Make your estimate.About 40°✓1
In plain EnglishAny estimate close to the real size (it is 38°) is good. Between 30° and 45° would be sensible.
Measuring and drawing angles
Use the protractor to measure the angle.(1 mark)
Line it up.Centre on the vertex, 0 along one arm
In plain EnglishPut the cross of the protractor on the vertex. Line up the baseline with the teal arm, so that arm points to 0.
Choose the right scale.The teal arm is at 0 on the inner scale
In plain EnglishThere are two scales. Use the one that starts at 0 on your arm. Here it's the inner (teal) scale.
Read where the other arm crosses.65°✓1
In plain EnglishCount up from 0 on the inner scale. The orange arm crosses at 65°. Sense check: it's acute, so 65° makes sense (not 115°).
⚠ Common mistake. Reading the wrong scale gives 115°. Always ask: is the angle acute or obtuse?
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Your turn
Type angles as numbers (you can leave out the ° sign). Stuck? Tap Need a hint?