# Angles

## Basics

### Key Idea

Angle measures a turn.

The angle between two line segments? measures how far one turns to go from one line to the other.

There are two main ways of measuring angles.

• Degrees KS4? and KS5?

When measuring in degrees, we effectively measure in fractions? of a full turn. Because it is easier to work with integers? rather than fractions?, we divide a full turn into 360 parts and count how many parts we turn. One degree is therefore one 360th of a full turn.

The symbol for degrees is ${}^\circ$.

When measuring in radians, we measure how far we travel when we travel around the circumference of a unit circle? when making the turn. There are therefore $2\pi$ radians in a full turn.

Radians are usually written without a symbol. When a symbol is used, it is ${}^c$.

As there are $360$ degrees and $2\pi$ radians in a full turn, the conversion is:

To ConvertMultiply By
Degrees to Radians$\frac{\pi}{180}$
Radians to Degrees$\frac{180}{\pi}$

## Results

All results about angles hinge on the following fact.

Angles are preserved by translation?, rotation?, reflection?, and scaling?.

### Basic Results

1. Angles around a point add up to $360^\circ$.

2. Angles at a point on a straight line add up to $180^\circ$.

3. Opposite angles are the same.

4. Corresponding angles are the same.

### Dependent Results

All of the following results can be deduced from the basic results.

1. Angles in a triangle? add up to $180^\circ$.

2. Alternate angles are the same.

3. Co-interior angles add up to $180^\circ$.

### Circle Theorems

The key to the circle theorems is the abundance of isosceles triangles. Any triangle? which is formed using two radii? of the same circle? must be isosceles?.

1. The angle in a semicircle? is a right angle?.

2. The angle at the centre is twice the angle at the circumference?.

3. Angles in the same segment? are equal.

category: geometry