Question

prove: p || q. we know that angle 1 is congruent to angle 3 and that line l is parallel to line m because. we see that is congruent to by the alternate interior angles theorem. therefore, angle 1 is congruent to angle 2 by the transitive property. so, we can conclude that lines p and q are parallel by the. x corresponding angles theorem, converse alternate exterior angles theorem, converse same side interior angles theorem

Explanation:

Step1: Identify given congruent angles

Given $\angle1\cong\angle3$.

Step2: Use alternate - interior angles theorem

Since $l\parallel m$, by the alternate - interior angles theorem, $\angle2\cong\angle3$.

Step3: Apply transitive property

Since $\angle1\cong\angle3$ and $\angle2\cong\angle3$, by the transitive property of congruence, $\angle1\cong\angle2$.

Step4: Use converse of corresponding angles theorem

If corresponding angles are congruent, then the lines are parallel. Since $\angle1$ and $\angle2$ are corresponding angles and $\angle1\cong\angle2$, we can conclude that $p\parallel q$ by the converse corresponding angles theorem.

Answer:

The reason for $l\parallel m$ is not given in the problem, but after showing $\angle1\cong\angle2$ (corresponding angles), we use the converse corresponding angles theorem to prove $p\parallel q$. So the correct choice is the converse corresponding angles theorem.