Exercise 22 Page 485

Let's consider the given information and the desired outcome of the proof. Let's write a paragraph proof! We begin by taking a look at the given diagram.

Notice that $\angle 2$ and $\angle 3$ are vertical angles. Therefore, by the Vertical Angles Theorem, we can say that these angles are congruent.

By the Vertical Angles Theorem, since $\angle 2$ and $\angle 3$ are vertical angles, they are congruent.

We are given that $\angle 1$ and $\angle 3$ are complementary angles and that $\angle 2$ and $\angle 4$ are also complementary angles. The Congruent Complements Theorem states that if two angles are complements of the same angle — or to congruent angles — then the two angles are congruent. Therefore, by this theorem, we can say that $\angle 1$ and $\angle 4$ are congruent angles.

By the Congruent Complements Theorem, $\angle 1$ and $\angle 4$ are congruent angles.

Completed Proof

Considering the given information, we can summarize all the steps in a paragraph proof. Proof: $\angle 2$ and $\angle 3$ are vertical angles. Therefore, by the Vertical Angles Theorem, $\angle 2$ and $\angle 3$ are congruent. By the Congruent Complements Theorem, $\angle 1$ and $\angle 4$ are congruent angles.