Question

writing a two - column proof
given: \\( \angle t \cong \angle v ; \overline{st} \parallel \overline{uv} \\)
prove: \\( \overline{tu} \parallel \overline{vw} \\)
complete the two - column proof.

Explanation:

Step1: Justify ∠T ≅ ∠U

Since \( \overline{ST}\parallel\overline{UV}\), by the Alternate - Interior Angles Theorem (if two parallel lines are cut by a transversal, then the alternate - interior angles are congruent), when the transversal is \( \overline{TU}\), we have \( \angle T\cong\angle U\).

Step2: Justify ∠U ≅ ∠V

We know that \( \angle T\cong\angle U\) (from step 1) and \( \angle T\cong\angle V\) (given). By the Transitive Property of Congruence (if \(a\cong b\) and \(a\cong c\), then \(b\cong c\)), we can conclude that \( \angle U\cong\angle V\).

Step3: Justify \( \overline{TU}\parallel\overline{VW}\)

Since \( \angle U\cong\angle V\), by the Converse of the Alternate - Interior Angles Theorem (if two lines are cut by a transversal such that alternate - interior angles are congruent, then the two lines are parallel), when the transversal is \( \overline{VU}\), we have \( \overline{TU}\parallel\overline{VW}\).

Answer:

1. Alternate - Interior Angles Theorem; 4. Transitive Property of Congruence; 5. Converse of the Alternate - Interior Angles Theorem