Question

proving parallel lines with alternate exterior angles
given: ∠1 ≅ ∠2
prove: p || q
determine the missing reasons in the two column proof of the converse of the alternate exterior angles the

Explanation:

Step1: Identify Vertical Angles

$\angle 1$ and $\angle 3$ are vertical angles, so $\angle 1 \cong \angle 3$ (Vertical Angles Theorem).

Step2: Transitive Property of Congruence

Given $\angle 1 \cong \angle 2$ (from Step 1) and $\angle 1 \cong \angle 3$ (from Step 2), by Transitive Property of Congruence, $\angle 2 \cong \angle 3$.

Step3: Converse of Alternate Exterior Angles

Since $\angle 2 \cong \angle 3$ (alternate exterior angles), then $p \parallel q$ (Converse of Alternate Exterior Angles Theorem).

Answer:

To complete the two - column proof:

• Statement 2: $\angle 1\cong\angle 3$; Reason 2: Vertical Angles Theorem.
• Statement 3: $\angle 2\cong\angle 3$; Reason 3: Transitive Property of Congruence.
• Statement 4: $p\parallel q$; Reason 4: Converse of Alternate Exterior Angles Theorem.