Question

Write a plan for a proof.

Given: $\angle \mathbf{3}$is a supplement of $\angle 1$;

$\angle 4$is a supplement of $\angle 2$.

Prove:$\angle \mathbf{3}\cong \angle \mathbf{4}$.

Short answer

Therefore, using$\angle 1$and$\angle 3$ and$\angle 2$ and $\angle 4$ are supplementary. It is proved that $\angle 3\cong \angle 4$ .

Step-by-step solution

Step 1. Observe the diagram.

∠3 is a supplement of ∠1 and ∠4 is a supplement of ∠2.

The diagram is:

Step 2. Show the proof.

Consider the following:

∠1 and∠3 and∠2 and∠4 are supplementary.

To prove $\angle 3\cong \angle 4$.

Since,∠1 and∠3 and∠2 and∠4 are supplementary.

Therefore,

$\angle 1+\angle 3={180}^{\circ }$

$\angle 2+\angle 4={180}^{\circ }$

Also, $\angle 1=\angle 2$both are vertically opposite angle.

Therefore,

$\angle 1+\angle 3={180}^{\circ }$

$\angle 1+\angle 4={180}^{\circ }$

Therefore, $\angle 3\cong \angle 4$(Hence proved)

Step 3. State the conclusion.

Therefore, using ∠1 and ∠3 and ∠2 and ∠4 are supplementary. It is proved that $\angle 3\cong \angle 4$.