Angles
Missing angles in triangles, on lines, round a point and in four-sided shapes, plus the angle facts to know by heart: which total to use, the quick way to the answer, and the traps the wrong answers set.
Chapter 1
What are angles questions?
An angle is the amount of turn between two lines that meet, measured in degrees (°). Each question gives you some angles and asks for a missing one.
You work it out from facts that are always true, not by measuring: the drawing is marked “not to scale”.
The shape. A triangle, a straight line, a point, crossing lines? It tells you which total to use.
The angles you know. On the drawing, or in the question.
A triangle has angles of 78° and 40°. What is the third angle?
The options
- Option 1: 140
- Option 2: 102
- Option 3: 118
- Option 4: 62
A Maths question from Rush, level 2.
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Turn, not length
An angle is how far one arm turns from the other. Longer arms don’t make a bigger angle: these two are both 40°.
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Acute Right Obtuse Reflex Names by size
Acute is under 90°. Right is exactly 90°, a quarter turn, marked with a little square. Obtuse is between 90° and 180°. Straight is 180°, a half turn. Reflex is over 180°. A full turn is 360°.
Chapter 2
Solve it in 4 steps.
Here’s the question, one step at a time.
The question
A triangle has angles of 78° and 40°. What is the third angle?
The options
- Option 1: 140
- Option 2: 102
- Option 3: 118
- Option 4: 62
A Maths question from Rush, level 2.
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Step 1
Pick the total.
Every missing angle rests on one of six facts. Find the one that matches the drawing. Here it’s a triangle: its three angles always make 180°.
- Right angle90°
- Straight line180°
- Round a point360°
- Triangle180°
- Quadrilateral360°
- Lines crossOpposites equal
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Step 2
Add the angles you know.
78 + 40 = 118. That’s the two together, not the third angle yet. 118 is one of the options, waiting for anyone who stops here.
78 + 40 = 118
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Step 3
Take it from the total.
180 − 118 is easiest counted up: 118 to 120 is 2, and 120 to 180 is 60. 2 + 60 = 62, so the missing angle is 62°.
2 + 60 = 62
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Step 4
Check it makes 180.
78 + 40 + 62 = 180. The check also catches an answer that’s ten out, a slip counting in tens: about three in four of Rush’s Angles questions offer one. And it’s why a triangle makes 180°: its three corners fit on a straight line.
The three corners fill a straight line: 180°.
Three mistakes it sets
| The mistake | Do this instead |
|---|---|
| Stopping at the two added. 118 is 78 + 40. | Take it from the total: 180 − 118 = 62. |
| Taking away only the 40. 140 is 180 − 40. | Add both first, then take them away together. |
| Taking away only the 78. 102 is 180 − 78. | Check: 78 + 40 + 102 is 220, too many. |
So it’s 62°. 78 + 40 + 62 = 180, the total for a triangle.
Remember: Total first, then take away.
Chapter 3
Everything else in the Angles game.
Each card is one more kind Rush asks: the fact, an example and its trap.
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A square corner or a straight line
A square corner makes 90°. A straight line makes 180°.
90 − 26 = 64°, and 180 − 115 = 65°. Traps: the angle you were given copied, and 25, from taking 115 from 90 instead of 180.
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Round a point: 360°
Angles all the way round a point make a full turn.
125 + 150 = 275, and 275 to 360 is 85°. Trap: 275, the two added, offered in more than half of Rush’s three-angle questions.
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Two lines cross
Vertically opposite angles are equal. Side by side, they make 180°.
Opposite 48° is 48°; next to it is 180 − 48 = 132°. Trap: each answer is the other one’s trap, so read which it asks for. A parallelogram’s angles work the same way.
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Isosceles: two angles match
Ticks mark two equal sides; the angles at their foot match.
Top angle 50°: 180 − 50 = 130, halved is 65° each. Trap: 130, not halved. Given base angles of 63°, take both away: 54°, not 117.
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Quadrilaterals: 360°
A quadrilateral (a four-sided shape) is two triangles: 360°.
70 + 90 + 140 = 300, so the fourth angle is 60°. Traps: 300, the three added, and 200, with the 140 left out.
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Angles written with x
Count the shares, then share out the total.
2x, 3x and 5x are 10 shares of 180°, so x = 18. Traps: 60, from sharing 180 equally between three, and 90, which is 5x.
Chapter 4
The angle facts in Shape Facts.
Shape Facts, Rush’s other game on this page, asks these four.
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A B C D Which angle is obtuse?
Hold each one up against a square corner and a straight line.
Only C is between 90° and 180°. Traps: B is a straight line, exactly 180°, and D’s arc goes the long way round, so it’s reflex.
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Turning to face a new way
Each quarter turn is 90°. Check which way you turn.
West, anticlockwise to south, is a quarter turn. Trap: clockwise it’s three quarters, and that’s offered too. North-east is halfway between north and east: 45°.
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Clock hands: 30° an hour
12 hours make 360°, so each hour is 30°.
At 4 o’clock the hands are 4 hours apart: 4 × 30 = 120°. Trap: 240°, the bigger angle, when it asks for the smaller.
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The angles inside a polygon
Cut it into triangles from one corner: two fewer than its sides.
A pentagon makes 3 triangles: 3 × 180 = 540°. Regular, all five match: 540 ÷ 5 = 108°. Trap: 900, one triangle for every side.
Chapter 5: In the exam
Two facts in one drawing.
Papers draw most angle questions, and often chain two facts in one.
Not to scale.
The triangle stands on a straight line, and the two marked sides are equal. The angle outside it is 110°. What is the angle at the top?
A 55°B 70°C 110°D 40°E 140°
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Step 1
Straight line first.
The 110° and the corner beside it make a straight line: 180 − 110 = 70°.
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Step 2
Isosceles: copy it across.
The ticked sides meet at the top, so the angles at their foot match: the other base angle is 70° too.
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Step 3
Triangle: take both away.
70 + 70 = 140, and 180 − 140 = 40°.
Three mistakes it sets
| The mistake | Do this instead |
|---|---|
| Stopping after one fact. 70° (B) is a base angle. | Keep going to the question mark. |
| Adding and stopping. 140° (E) is 70 + 70. | Take it from 180: 180 − 140 = 40. |
| Halving the wrong angle. 55° (A) is (180 − 70) ÷ 2. | The 70° is a base angle, not the top. |
So it’s 40°, D. The line leaves 70°, the ticks copy it, and 180 − 140 = 40.
Remember: One fact at a time.
Other ways the exam asks it
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The outside angle shortcut
The angle outside equals the two inside angles far from it.
35 + 80 = 115°. The long way agrees: 180 − 115 = 65 inside, and 35 + 80 + 65 = 180. A shortcut, not something school expects in Year 6.
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Measuring with a protractor
Read the scale that starts at 0 on the arm you line up.
Some papers ask you to measure (the Buckinghamshire test doesn’t allow a protractor). This one is acute, so it’s 50°, not 130°: the trap from the other scale.
Chapter 6
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- Pick the total.
- Add the angles you know.
- Take it from the total.
- Check it makes 180.
Remember: Total first, then take away.
For parents.
What parents ask us about angles.
Questions parents ask
What angle facts do you need for the 11+?
Angles on a straight line make 180°, round a point 360°, in a triangle 180° and in a quadrilateral 360°. Vertically opposite angles, where two lines cross, are equal. An isosceles triangle has two equal angles, and an equilateral triangle has three angles of 60°. All of these are on the Year 5 and Year 6 curriculum.
How do you find a missing angle in a triangle?
Add the two angles you know and take the answer from 180°. For 92° and 51°: 92 + 51 = 143, and 180 − 143 = 37°. Then check that all three add to exactly 180.
Why do the angles in a triangle add up to 180°?
Tear the three corners off a paper triangle and line them up point to point: they make a straight line, and a straight line is 180°. It works with any triangle.
Are 11+ angle diagrams drawn to scale?
Usually not, and they say so. Work from the numbers, not by measuring or judging by eye, unless the question tells you to measure.
Rush: 11+ Practice & KS2 SATsPractise angles in Rush.
- Angles and Shape Facts as 100-second games, easy at first and harder as your child improves
- Every miss kept with the right answer and why
- Angles and Shape Facts are in Premium, and six other games are free
On the App Store for iPhone and iPad
Six other games are free, plus a new family game every day. Angles and Shape Facts are in Premium: all 41 games are £24.99 a year after a free week1, or £3.99 a month.