# Angle

## Definition

An angle is the union of two rays with a common endpoint. The common endpoint of the rays is called the vertex of the angle, and the rays themselves are called the sides of the angle. A ray drawn from the vertex of the angle, such that the angle formed by this ray and one of the sides is congruent to the angle formed by this ray and the other side, is called the angle bisector.

There are many notations for angles. The most common form is $\angle ABC$, read "angle ABC", where $A,C$ are points on the sides of the angle and $B$ is the vertex of the angle. Note that the same angle can be denoted many different ways by choosing different points along the side of the angle. If there is no ambiguity, this notation can be shortened to simply $\angle B$.

## Angle Measure

The measure of $\angle ABC$ is denoted $m\angle ABC$, read "measure of angle ABC". There are different units for measuring angles. The three most common are degrees, radians and gradians.

Congruent angles have the same angle measure.

## Special Angles

### Straight Angle

A straight angle is an angle formed by a pair of opposite rays, or a line. A straight angle has a measure of $180^\circ=\pi\; \mathrm{ rad}$.

### Right Angle

A right angle is an angle that is supplementary to itself. A right angle has a measure of $90^\circ=\frac{\pi}{2}\;\mathrm{ rad}$.

An acute angle has a measure greater than zero but less than that of a right angle, i.e. $\angle ABC$ is acute $\Leftrightarrow 0^\circ.

An obtuse angle has a measure greater than that of a right angle but less than that of a straight angle, i.e. $\angle ABC$ is obtuse $\Leftrightarrow 90^\circ.

### Reflex Angle

A reflex angle is an angle with measure greater than a straight angle, but less than 360 degrees, or $2\pi$ radians.