Definition:Complementary
Definition
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.
Another (equivalent) definition is to say that two angles, which, when set next to each other, form a right angle are complementary.
Linguistic Note
The word "complement" comes from the idea of "complete-ment", it being the angle needed to "complete" a right angle.
It is a common mistake to confuse the words "complement" and "compliment". Usually the latter is mistakenly used when the former is meant.
