Question

1. prove the corresponding angles theorem. given: m || n prove: ∠1 ≅ ∠2

Explanation:

Step1: Identify vertical - angle relationship

$\angle1$ and $\angle3$ are vertical angles, so $\angle1\cong\angle3$ (Vertical angles are congruent).

Step2: Identify alternate - interior angle relationship

Since $m\parallel n$, $\angle3$ and $\angle2$ are alternate - interior angles. So $\angle3\cong\angle2$ (Alternate - interior angles of parallel lines are congruent).

Step3: Use the transitive property

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

Answer:

$\angle1\cong\angle2$ is proven using the properties of vertical angles, alternate - interior angles of parallel lines, and the transitive property of congruence.