Question

Describe your plan for proving the following. You don’t need to give all the details.

Given: $\overline{AC}\perp \overline{BC}$;∠3 is complementary to ∠1.

Prove: $\angle 3\cong \angle 2$

Short answer

It is proven that$\angle 3\cong \angle 2$.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Definition of complementary angles.

The complementary angles are the angles whose sum is 90°.

Step 3. Write the proof of the statement ∠3≅∠2.

∠3 is complementary to∠1.

Therefore, the sum of the angles∠3 and∠1 is 90°.

That implies,

$\angle 3+\angle 1=90°\left(1\right)$

As$\overline{AC}\perp \overline{BC}$, therefore the lines AC and BC are perpendicular and therefore by using the definition of the perpendicular lines the measure of the angle $\angle ACB=90°$.

By using the angle addition postulate:

$\angle ACB=\angle 1+\angle 2$.

Now,

$\begin{array}{c}\angle ACB=90°\\ \angle 1+\angle 2=90°\left(2\right)\end{array}$

Subtract the equation (2) from (1).

$$\begin{gathered} \angle 3+\angle 1-\left(\angle 1+\angle 2\right)=90°-90° \\ \angle 3+\angle 1-\angle 1-\angle 2=0 \\ \angle 3-\angle 2=0 \\ \angle 3=\angle 2 \end{gathered}$$

Therefore, $\angle 3\cong \angle 2$.