Definition:Complement

Geometry


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.

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.