Question

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

Given:∠3 is supplementary to∠1;∠4 is supplementary to∠2.

Prove: $\angle 3\cong \angle 4$

Short answer

∠3 and ∠4 are equivalent.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Definition of supplementary angles.

The supplementary angles are the angles whose sum is 180°.

Step 3. Write the proof of the statement ∠3≅∠4.

∠3 is supplementary to∠1 and∠4 is supplementary to∠2.

Therefore, the sum of angles ∠3 and ∠1 is 180° and the sum of angles ∠4 and ∠2 is also 180°.

That implies,

$\angle 3+\angle 1=180°\left(1\right)$and $\angle 4+\angle 2=180°\left(2\right)$.

Subtract equation (2) from equation (1).

$\begin{array}{c}\angle 3+\angle 1-\left(\angle 4+\angle 2\right)=180°-180°\\ \angle 3+\angle 1-\angle 4-\angle 2=0\left(3\right)\end{array}$

From the given diagram it can be noticed that angles ∠1 and ∠2 are the vertical angles.

As, angles ∠1 and ∠2 are congruent angles, therefore $\angle 1=\angle 2$.

Substitute the above relation in equation (3).

$$\begin{gathered} \angle 3+\angle 1-\angle 4-\angle 2=0 \\ \angle 3+\angle 2-\angle 4-\angle 2=0 \\ \angle 3-\angle 4=0 \\ \angle 3=\angle 4 \end{gathered}$$

Therefore, $\angle 3\cong \angle 4$.