Chapter 2.6, Problem 9CE

To determine

To describe: Plan for proving $\angle 3\cong \angle 2$

Explanation of Solution

Given:

The given figure is,

  

And the condition is,

  $\overleftrightarrow{AC}\perp \overleftrightarrow{BC}$ and

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

As given,

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

So, $m\angle 3+m\angle 1=90$ ...... (1)

Also, $\overleftrightarrow{AC}\perp \overleftrightarrow{BC}$

So,

  $\begin{array}{l}\angle ACB=90\\ \Rightarrow m\angle 1+m\angle 2=90\mathrm{.....}(2)\end{array}$

From (1) and (2)

  $\begin{array}{l}m\angle 3+m\angle 1=m\angle 1+m\angle 2\\ \Rightarrow m\angle 3=m\angle 2\end{array}$

Hence, $\angle 3\cong \angle 2$