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.

• An acute angle is less than $90\degree$ — a sharp little corner.
• A right angle is exactly $90\degree$ — a perfect square corner, often shown with a small square.
• An obtuse angle is between $90\degree$ and $180\degree$.
• A straight angle is exactly $180\degree$ — a flat half-turn.
• A reflex angle is bigger than $180\degree$ but less than $360\degree$.

Knowing the name gives you a quick check: if you measure an obtuse angle and get $40\degree$, you know something has gone wrong.

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 $65\degree$ on the matching scale, then join the vertex to your dot.

Always check your answer makes sense. If the angle clearly looks acute, your reading must be under $90\degree$.

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 $x = 60\degree$.

Angles that meet all the way around a point make a full turn, so they add up to $360\degree$.

If three angles around a point are $150\degree$, $130\degree$ and $y$, then: $y = 360\degree - 150\degree - 130\degree = 80\degree.$

The method is always the same: write down the total the angles must reach, then subtract the ones you know.

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 $70\degree$, and the other two are each $180\degree - 70\degree = 110\degree$.

Watch your words: vertically opposite does not mean "up and down" — it means the angle on the opposite side of the vertex. A pair can sit side by side and still be vertically opposite.

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: $x = 180\degree - 55\degree - 80\degree = 45\degree.$

Some triangles give you extra clues:

• An equilateral triangle has three equal sides and three equal angles of $60\degree$ each.
• An isosceles triangle has two equal sides and the two base angles are equal.
• A right-angled triangle has one $90\degree$ angle, so the other two must add up to $90\degree$.

If you ever get a triangle answer above $180\degree$, or a "third angle" of $0\degree$ or less, you know to check your working.

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 $180\degree$ (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.