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.

Linguistic Note

The word complement comes from the idea of complete-ment, it being the thing needed to complete something else.

It is a common mistake to confuse the words complement and compliment. Usually the latter is mistakenly used when the former is meant.

Also see