Angles

An angle measures how far you turn. Open a door a little and you make a small angle; swing it wide open and the angle is bigger. We measure angles in degrees, written with a small circle: a quarter turn is $90\degree$, a full spin is $360\degree$.

Angles are grouped by size, and each group has its own name.

A protractor is the tool for measuring and drawing angles. It has two scales of numbers — one running clockwise, one running anticlockwise — so the trick is choosing the correct one.

To measure an angle:

1. Put the protractor's centre cross exactly on the vertex (the point of the angle).
2. Line one arm of the angle up with the $0\degree$ line.
3. Read the scale that starts at zero on that arm, following it round to the other arm.

To draw an angle, say $65\degree$: draw one arm, place the protractor centre on the end, mark a dot at on the matching scale, then join the vertex to your dot.

Some angle facts come straight from the idea of a turn, and they let you find a missing angle by calculating instead of measuring.

Angles that sit together on a straight line make a half-turn, so they add up to $180\degree$.

So in the diagram, $x + 120\degree = 180\degree$, which gives .

When two straight lines cross, they make four angles. The angles that sit opposite each other at the crossing point are always equal — these are called vertically opposite angles.

This makes sense from the straight-line rule. Each pair of angles next to each other lies on a straight line, so they add up to $180\degree$. That forces the opposite angles to match.

So if one angle at a crossing is $70\degree$, the angle directly opposite is also , and the other two are each .

Here is one of the most useful facts in all of geometry: the three angles inside any triangle add up to $180\degree$. It does not matter whether the triangle is big, small, thin or wide.

To find a missing angle, add the two you know and subtract from $180\degree$. In the diagram: $x=180\mathrm{°}-55$

Parallel lines are lines that stay the same distance apart and never meet — shown with little matching arrows. When another straight line (a transversal) crosses a pair of parallel lines, special equal angles appear.

Three pairs are worth knowing — each has a letter shape to help you spot it:

• Corresponding angles (the F shape) are in matching corners and are equal.
• Alternate angles (the Z shape) are on opposite sides of the transversal and are equal.
• Co-interior angles (the C shape) are between the parallel lines and add up to $180\degree$.

To find an unknown angle, hunt for the F, Z or C shape, then either copy the angle (F and Z) or subtract from (C). You can often chain these rules together with the straight-line and vertically-opposite facts to solve a whole diagram.

Getting confident with angles now opens up a lot of geometry to come:

• Polygons — finding the angle sum of quadrilaterals, pentagons and beyond all builds on the triangle rule.
• Constructions and bearings — measuring and drawing angles accurately is essential for scale drawings and navigation.
• Transformations — describing a rotation needs an angle of turn.
• In everyday life, angles appear in design, sport, building, ramps and even reading a clock.

If a later geometry topic feels tricky, it is often an angle fact underneath that needs a quick refresh. Keep this guide handy — there is no rush, and every diagram gets easier with practice.