Prove that sum of two complementary angles is ${90}^{\circ }$.

If two angles are complementary, that means that their sum is ${90}^{\circ }$.
Two angles form a pair of complementary, supplementary and vertical angles. When the sum of two angles is ${90}^{\circ }$, they are called complementary angles. Such a relationship may be useful in determining other unknown angles.


Complementary Angle Theorem
Statement: If two angles are each complementary to a 3rd angle, then they are congruent to each other. Or complements of the same angle are congruent.
Proof: Let $\angle A$ and $\angle B$ be complementary angles.
Step 1: $\angle B$ is a complement of $\angle A$
Then $\angle A+\angle B={90}^{\circ }$
(By complementary angles definition)
Step 2: Let $\angle A$ and $\angle C$ be complementary angles.
$\Rightarrow \angle C$ is a complement of $\angle A$
Then, $\angle A+\angle C={90}^{\circ }$
From Step 1 and Step 2 we have,
$\angle A+\angle B=\angle A+\angle C$
$\Rightarrow \angle B=\angle C$