Question

in the figure below, lines m and n are cut by transversal t. which of the following is a valid step in the proof that corresponding angles ∠1 and ∠2 are congruent? show your work here ∠3 ≅ ∠4 because same - side interior angles are congruent ∠2 ≅ ∠3 because vertical angles are congruent ∠1 ≅ ∠4 because adjacent angles are congruent m∠3 + m∠4 = 180° because same - side interior angles are supplementary

Explanation:

Step1: Recall angle - relationship properties

We know different angle - relationship rules for lines cut by a transversal.

Step2: Analyze each option

• Option 1: Same - side interior angles are supplementary, not congruent, so $\angle3\cong\angle4$ is wrong.
• Option 2: $\angle2$ and $\angle3$ are vertical angles. By the vertical - angles theorem, vertical angles are congruent. This is a valid step in the proof.
• Option 3: Adjacent angles are not necessarily congruent. $\angle1$ and $\angle4$ are adjacent angles and $\angle1\cong\angle4$ is wrong.
• Option 4: While $m\angle3 + m\angle4=180^{\circ}$ is true for same - side interior angles, it is not a step to prove $\angle1\cong\angle2$.

Answer:

$\angle2\cong\angle3$ because vertical angles are congruent