Definition:Complementary Angles



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 $$\frac \pi 2 - \alpha$$.

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 $$\frac{\pi}{2}$$ 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.

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