Angles Explained: Types of Angles and Angle Facts

Q: What is a reflex angle?

An angle bigger than 180° but less than 360°: more than a straight line, less than a full turn.

Q: Why are vertically opposite angles equal?

Each one makes 180° with the same neighbouring angle on a straight line, so they must be equal to each other.

What an angle measures, the names for acute, right, obtuse and reflex angles, and the three angle facts (on a line, around a point and vertically opposite) that solve most angle problems.

3 min read Supports 3 lessons

An angle measures how far something has turned. Open a door a little and the angle is small; swing it all the way back against the wall and the angle is large.

We measure angles in degrees. A complete turn is split into $360$ equal degrees, a number chosen by ancient astronomers because it divides so many ways.

Interactive

130°

obtuse angle

On a straight line it pairs with 50°

Around a point it leaves 230°

Angle130°

Names for angles

Tip

Acute $< 90°$ $<$ obtuse $< 180°$ $<$ reflex $< 360°$ . Exactly $90°$ is a right angle, marked with a little square; exactly $180°$ is a straight angle.

The three angle facts

Angle facts

\text{on a line: } 180^\circ \qquad \text{around a point: } 360^\circ \qquad \text{vertically opposite: equal}

On a straight line. A straight line is half a turn, so angles that sit together on one side of a line add to $180°$ .

Around a point. Angles that go all the way round a point make a full turn, so they add to $360°$ .

Vertically opposite. When two straight lines cross, the angles opposite each other are equal. Each pair of neighbours makes $180°$ , so opposite angles must match.

$\text{Two lines cross. One angle is } 65^\circ$
Find the other three.

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Worked example

Three angles on a straight line are $x$ , $2x$ and $45°$ . Find $x$ .

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They add to $180°$ .
$x + 2x + 45 = 180$

Parallel lines

When a line crosses two parallel lines, the same angles appear at both crossings. Matching positions give corresponding angles (equal); the inner zig-zag gives alternate angles (equal); and the two inner angles on the same side add to $180°$ .

Key idea

Angle problems are usually solved with a chain of these small facts. Give a reason for each step and the answer follows.

Beyond degrees

Degrees are convenient but arbitrary. Calculus measures angles in radians, where a full turn is $2\pi$ .

Common questions

What is a reflex angle?: An angle bigger than $180°$ but less than $360°$ : more than a straight line, less than a full turn.
Why are vertically opposite angles equal?: Each one makes $180°$ with the same neighbouring angle on a straight line, so they must be equal to each other.

Practise angles

The Angles lesson has worked examples and unlimited practice that adapts to you.

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Common questions

What is a reflex angle?

An angle bigger than $180°$ but less than $360°$ : more than a straight line, less than a full turn.

Why are vertically opposite angles equal?

Each one makes $180°$ with the same neighbouring angle on a straight line, so they must be equal to each other.