Year 10 Foundation · Angles
Work out angles around a point, on a line and where lines cross
Learn how to work out missing angles using facts about points, lines, triangles, polygons and parallel lines, and set out your reasons as a proof.
Definition
An angle measures a turn between two lines, in degrees. A small set of facts, such as angles on a straight line adding to 180° and angles around a point adding to 360°, lets you calculate missing angles instead of measuring them. At GCSE the working matters as much as the answer: each step should name the angle fact that justifies it, and a fully reasoned chain of steps is a proof.
Why Do I Have to Learn This?
Angles measure turn, from door hinges to skate ramps. Reading them helps you build, draw and navigate accurately.
Key Rules:
- 1. Angles around a point add to 360°, angles on a straight line add to 180°, and vertically opposite angles, made where two lines cross, are equal.
Remember:
- •Point 360°, line 180°, triangle 180°, quadrilateral 360°. Name the fact you use at every step.
Don’t Forget...
- •Using 180° for angles around a point, or 360° for angles on a straight line. A straight line is only half a full turn, so angles on a line total 180°, and it takes a complete turn around a point to make 360°.
- •Dividing 360° by the number of sides to get the interior angle of a regular polygon. 360° divided by the number of sides gives the exterior angle, and the interior angle is 180° minus that, so a regular hexagon has interior angles of 120°, not 60°.
- •Calling co-interior angles equal because the lines are parallel. Between parallel lines only alternate and corresponding angles are equal; co-interior angles add to 180°, so an angle of 110° pairs with 70°, not another 110°.
Acute angle
An angle smaller than 90 degrees.
An angle of 45° is acute.
Worked Example:
In triangle ABC, angle A is 52° and angle B is 63°. The side BC is extended beyond C to a point D. Work out the size of angle ACD, giving a reason for each step.
Angles in a triangle add to 180°, so angle ACB = 180° - 52° - 63° = 65°.
BCD is a straight line, so angle ACB and angle ACD add to 180°.
Angle ACD = 180° - 65° = 115°.
Check: the exterior angle 115° equals 52° + 63°, the sum of the two interior angles furthest from it.
Answer: Angle ACD = 115°
Why It Works:
A full turn is 360° and a straight line is half a turn, so angles on a line total 180°; every fact in this unit unfolds from those two. For example, walking once around the outside of any polygon turns you through exactly one full turn, which is why the exterior angles always add to 360°.
Another Example:
Work out the size of each interior angle of a regular octagon.
The exterior angles of any polygon add to 360°, and a regular octagon has 8 equal ones.
Each exterior angle is 360° ÷ 8 = 45°.
The interior and exterior angle at each corner sit together on a straight line, so they add to 180°.
Each interior angle is 180° - 45° = 135°.
Answer: Each interior angle is 135°
Animated Example:
Free Work out angles around a point, on a line and where lines cross Worksheet
Download free worksheets and practise today.
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Let’s Practise the Concept
Step 1 of 3
Put the angles in order
Tap them from smallest to largest.
In order so far: nothing yet