Question

1. fill in the missing steps of the proof of the converse of the alternate interior angles theorem. given: ∠1≅∠2 prove: a∥b statements: 1) ∠1≅∠2 2) 3) ∠2≅∠3 4) a∥b reasons: 1) given 2) 3) vertical angles theorem 4)

Explanation:

Step1: Given information

We are given that $\angle1\cong\angle2$.

Step2: Use vertical - angle property

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

Step3: Transitive property of congruence

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

Step4: Converse of alternate - interior angles theorem

If $\angle1\cong\angle3$ (alternate - interior angles), then $a\parallel b$ by the converse of the alternate - interior angles theorem.

Answer:

1. $\angle1\cong\angle3$; Transitive property of congruence
2. $\angle1$ and $\angle3$ are alternate - interior angles; Definition of alternate - interior angles
3. If alternate - interior angles are congruent, then lines are parallel; Converse of alternate - interior angles theorem