Chapter 2.6, Problem 18WE

To determine

To prove: angles complement of congruent angles are congruent.

Explanation of Solution

Given information: Two angles that are complements of congruent angles.

Formula used: Sum of two complementary angles is $90°$ .

Proof:

Consider the figure shown.

  

Given:

  $\angle 1$ is complementary to $\angle 3$

  $\angle 2$ is complementary to $\angle 4$

  $\angle 3\cong \angle 4$

To prove: $\angle 1\cong \angle 2$

Proof:

As $\angle 1$ is complementary to $\angle 3$ ,

  $\angle 1+\angle 3=90°(i)$

(Sum of two complementary angles is $90°$ )

As $\angle 2$ is complementary to $\angle 4$

  $\angle 2+\angle 4=90°(ii)$

(Sum of two complementary angles is $90°$ )

Subtract $(i)$ from $(ii)$

  $\begin{array}{l}\left(\angle 2+\angle 4\right)-\left(\angle 1+\angle 3\right)=90°-90°\\ \Rightarrow \left(\angle 2+\angle 4\right)-\left(\angle 1+\angle 3\right)=0(iii)\end{array}$

Also,

  $\begin{array}{l}\angle 3\cong \angle 4\\ \Rightarrow \angle 3=\angle 4(iv)\end{array}$

(if two angles are congruent then they are equal to each other)

So,

  $\begin{array}{l}\angle 2-\angle 1=0\\ \Rightarrow \angle 2=\angle 1\end{array}$

This implies $\angle 1\cong \angle 2$

(If two angles are equal then they are also congruent)