Exercise 26 Page 176

Let's begin by looking at the given information and the desired outcome of the proof. Then we can write a proof! Since $\angle C$ is a right angle, we can conclude that $m\angle C=90.$ Recall that the Triangle Angle-Sum Theorem tells us that the measures of the angles of a triangle is $180.$ Therefore, the sum of the measures of $\angle A,$ $\angle B,$ and $\angle C$ is $180.$ Now we have the following equations. By the Substitution Property of Equality, we can substitute $90$ for $m\angle C$ in the second equation. We got that the sum of angles $A$ and $B$ is $90.$ Therefore, by the definition of complementary angles we can conclude that $\angle A$ and $\angle B$ are complementary.

Final Proof

Prove: By the definition of a right angle, $m\angle C=90.$ By the Triangle Angle-Sum Theorem, $m\angle A+m\angle B+m\angle C=190.$ By the Substitution Property of Equality, $m\angle A+m\angle B=90.$ By the definition of complementary angles, $\angle A$ and $\angle B$ are complementary.