\begin{definition}[Definition:Complementary Angles]
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Let $\angle BAC$ be a right angle.
Let $\angle BAD + \angle DAC = \angle BAC$.
That is, $\angle DAC = \angle BAC - \angle BAD$.
Then $\angle DAC$ is the '''complement''' of $\angle BAD$.
Hence, for any angle $\alpha$ (whether less than a right angle or not), the complement of $\alpha$ is $\dfrac \pi 2 - \alpha$.
Measured in degrees, the complement of $\alpha$ is $90^\circ - \alpha$.
If $\alpha$ is the complement of $\beta$, then it follows that $\beta$ is the complement of $\alpha$.
Hence we can say that $\alpha$ and $\beta$ are '''complementary'''.
It can be seen from this that the '''complement''' of an angle greater than a right angle is negative.
Thus '''complementary angles''' are two angles whose measures add up to the measure of a right angle.
That is, their measurements add up to $90$ degrees or $\dfrac \pi 2$ radians.
\end{definition}