Year 10 Higher · Angles
Work out angles around a point, on a line and where lines cross
Learn how to work out angles on lines, in triangles and polygons and between parallel lines, and how to set out an angle proof.
Definition
An angle measures an amount of turn between two lines, in degrees. A small set of facts, such as angles on a straight line adding to 180°, fixes what every missing angle must be, so you can calculate angles exactly rather than measuring them. At Higher tier you are also expected to quote these facts as reasons, because a chain of justified statements is what makes an argument 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 on a straight line add to 180°, angles around a point add to 360°, and where two lines cross the vertically opposite angles are equal.
Remember:
- •Every missing angle comes from a named fact: quote the fact, then do the arithmetic.
Don’t Forget...
- •Treating co-interior angles as equal because alternate and corresponding angles are. Co-interior angles sit on the same side of the crossing line and add to 180°; they are only equal in the special case where both are 90°.
- •Using (n - 2) x 180° as the size of one interior angle instead of the total. The formula gives the sum of all the interior angles; for one angle of a regular polygon you must divide that total by the number of sides.
- •Writing correct statements in a proof without quoting the angle fact that justifies each one. A proof is marked on its reasons: a statement with no named fact behind it is an observation, and the chain of justification is broken.
Acute angle
An angle smaller than 90 degrees.
An angle of 45° is acute.
Worked Example:
A regular polygon has 10 sides. Work out the size of one interior angle.
The exterior angles of any polygon add to 360°.
The polygon is regular, so its 10 exterior angles are equal: 360 ÷ 10 = 36°.
At each vertex the interior angle and the exterior angle lie on a straight line, so they add to 180°.
One interior angle is 180 - 36 = 144°.
Answer: 144°
Why It Works:
Walking once around the boundary of any polygon turns you through exactly one full turn, which is why the exterior angles always total 360° whatever the number of sides. The interior total follows because each interior angle sits on a straight line with its exterior angle, so all n pairs contribute 180n degrees, and removing the 360° of exterior turn leaves (n - 2) x 180°.
Another Example:
A straight line crosses two parallel lines. Two co-interior angles measure 4x and x + 30. Work out the size of the larger angle, giving a reason for the equation you form.
Co-interior angles between parallel lines add to 180°, so 4x + x + 30 = 180.
Simplify and solve: 5x + 30 = 180, so 5x = 150 and x = 30.
Substitute back: the angles are 4 x 30 = 120° and 30 + 30 = 60°.
Check the reason still holds: 120 + 60 = 180, as co-interior angles must.
Answer: The larger angle is 120°
Animated Example:
Free Work out angles around a point, on a line and where lines cross Worksheet
Download free worksheets and practise today.
Every download has a fresh set of questions.
Before this: Angles in parallel lines and polygons (Year 8) · Next: Circle theorems (Year 11 Higher)
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