Angles & Lines (Years 6–9) — Maths

An angle measures the amount of turn between two lines, counted in degrees (°) — a full spin all the way round is 360°. When you open a door, steer a bike or read a clock hand sweeping round, you are watching angles change. A handful of simple facts let you find missing angles in almost any diagram without measuring.

Why this matters: angle facts are the toolkit behind carpentry, navigation, sport and design — a snooker player, a satellite dish installer and a roof builder are all really solving angle problems. In exams, a whole diagram usually unlocks from one known angle by chaining these rules together, so knowing them cold is what turns a hard-looking picture into a few subtractions.

Angles on a straight line add up to 180° (a + b = 180°)

The angle facts to know

Angles around a point make one full turn: 360°.

These four are the backbone. Each one is really the same idea — angles that fit together into a known total:

Angles on a straight line add up to 180° (a half turn).
Angles around a point add up to 360° (a full turn).
Angles in a triangle add up to 180°.
Angles in a quadrilateral (4-sided shape) add up to 360°.

To find a missing angle, add up the ones you know and take them from the total. The picture shows two angles on a line: since they fill a half turn, a + b = 180°.

Two angles on a line are 110° and x: they must total 180°, so x = 180 − 110 = 70°. A triangle has 50° and 60°: the third is 180 − 50 − 60 = 70°.

Parallel lines and their letter-shapes

Arrows mark the parallel lines; the Z-angles are equal.

When one straight line (a "transversal") crosses two parallel lines, it creates special pairs of related angles. The trick is to spot the letter shape traced by the lines — like reading the picture as a hidden alphabet:

Corresponding angles make an F-shape — they are equal.
Alternate angles make a Z-shape — they are equal.
Co-interior angles make a C-shape — they add up to 180°.

Rule	Total
On a straight line	180°
Around a point	360°
Angles in a triangle	180°
Angles in a quadrilateral	360°
Corresponding (F) angles	equal
Alternate (Z) angles	equal
Co-interior (C) angles	180°

An isosceles triangle with an apex of 40°: the two base angles are equal, so they share the remaining 180 − 40 = 140°, giving 140 ÷ 2 = 70° each.

Equal sides give equal base angles: (180 − 40) ÷ 2 = 70°.

Common mistake: muddling the two "equal" rules with the "add to 180°" one. F and Z angles are equal; only C (co-interior) angles are supplementary (sum 180°). If a C-shape answer comes out equal to the given angle, you have used the wrong rule.

Memory hook: Line 180, Point 360, Triangle 180, Quad 360. With parallel lines, hunt for the F, Z and C — F and Z are equal, C sums to 180°.

When the angles are written with a letter

Sometimes an angle is given as an expression like (3x − 20). Nothing about the rule changes — the angles still add to the same total. Write that total as an equation and solve it.

Worked example. Angles on a straight line are (3x − 20), (x + 40) and (2x − 8). On a straight line they must add to 180°:

(3x − 20) + (x + 40) + (2x − 8) = 180
Collect the terms: 6x + 12 = 180
Take 12 from both sides: 6x = 168
Divide by 6: x = 28

Always check. 3(28) − 20 = 64, 28 + 40 = 68, 2(28) − 8 = 48. And 64 + 68 + 48 = 180. ✓