Question

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

Given: $\angle 2\cong \angle 3$

Prove: $\angle 1\cong \angle 4$

Short answer

As, $\angle 2\cong \angle 3$,therefore the angles∠2and∠3are congruent angles.

Therefore, $\angle 2=\angle 3\left(1\right)$.

From the diagram it can be noticed that the angles∠2 and∠3 are the vertical angles.

Therefore, by using the theorem of vertical angles it can be said that the angles ∠1 and∠2are the congruent angles.

That implies,$\angle 1=\angle 2\left(2\right)$

From the diagram it can also be noticed that the angles∠3 and∠4are the vertical angles.

Therefore, by using the theorem of vertical angles it can be said that the angles ∠3 and∠4 are the congruent angles.

That implies,$\angle 3=\angle 4\left(3\right)$

By using the equations (1), (2) and (3):

$\begin{array}{l}\angle 2=\angle 3\\ \angle 1=\angle 3\left(\because \angle 1=\angle 2\right)\\ \angle 1=\angle 4\left(\because \angle 3=\angle 4\right)\end{array}$

Therefore, the angles∠1 and∠4have same measure.

Therefore, the angles∠1 and∠4 are the congruent angles.

That implies, $\angle 1\cong \angle 4$.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Write the theorem of vertical angles.

The theorem of vertical angles states that if the two angles are vertical then the angles are congruent.

Step 3. Write the proof of the statement ∠1≅∠4.

As, $\angle 2\cong \angle 3$, therefore angles ∠2 and ∠3 are congruent angles.

Therefore, $\angle 2=\angle 3\left(1\right)$.

From the diagram, it can be noticed that angles ∠2 and ∠3 are the vertical angles.

Therefore, by using the theorem of vertical angles it can be said that angles ∠1 and ∠2 are congruent angles.

That implies, $\angle 1=\angle 2\left(2\right)$

From the diagram, it can also be noticed that angles ∠3 and ∠4 are the vertical angles.

Therefore, by using the theorem of vertical angles it can be said that angles ∠3 and ∠4 are congruent angles.

That implies, role="math" localid="1646548206500" $\angle 3=\angle 4\left(3\right)$

By using the equations (1), (2) and (3):

$\begin{array}{l}\angle 2=\angle 3\\ \angle 1=\angle 3\left(\because \angle 1=\angle 2\right)\\ \angle 1=\angle 4\left(\because \angle 3=\angle 4\right)\end{array}$

Therefore, angles ∠1 and ∠4 have the same measure.

Therefore, angles ∠1 and ∠4 are congruent angles.

That implies, $\angle 1\cong \angle 4$.