# Definition:Supplementary Angles

## Definition

Let $\angle ACB$ be a straight angle.

Let $\angle BCD + \angle DCA = \angle ACB$.

That is, $\angle DCA = \angle ACB - \angle BCD$.

Then $\angle DCA$ is the **supplement** of $\angle BCD$.

Hence, for any angle $\alpha$ (whether less than a straight angle or not), the **supplement** of $\alpha$ is $\pi - \alpha$.

Measured in degrees, the **supplement** of $\alpha$ is $180^\circ - \alpha$.

If $\alpha$ is the **supplement** of $\beta$, then it follows that $\beta$ is the **supplement** of $\alpha$.

Hence we can say that $\alpha$ and $\beta$ are **supplementary**.

It can be seen from this that the **supplement** of a reflex angle is negative.

Thus, **supplementary angles** are two angles whose measures add up to the measure of $2$ right angles.

That is, their measurements add up to $180$ degrees or $\pi$ radians.

Another (equivalent) definition is to say that two angles are **supplementary** which, when set next to each other, form a straight angle.

## Also known as

Some sources use the term **adjacent angles** for this concept, but $\mathsf{Pr} \infty \mathsf{fWiki}$ has a different definition for that term.

## Also see

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**supplementary angles**