Question

given: \\( \angle t = \angle v ; \overline { s t } \parallel \overline { u v } \\) prove: \\( \overline { t u } \parallel \overline { v w } \\) complete the two - column proof.

Explanation:

• For statement 2 ($\angle T=\angle U$), since $\overline{ST}\parallel\overline{UV}$, by the Alternate - Interior Angles Theorem (when two parallel lines are cut by a transversal, alternate - interior angles are congruent), here the transversal is $\overline{TU}$.
• For statement 4 ($\angle U = \angle V$), using the Transitive Property of Congruence. If $\angle T=\angle U$ (from statement 2) and $\angle T=\angle V$ (from statement 3, given), then $\angle U=\angle V$.
• For statement 5 ($\overline{TU}\parallel\overline{VW}$), by the Converse of the Alternate - Interior Angles Theorem. If two lines are cut by a transversal and the alternate - interior angles are congruent, then the two lines are parallel. Here, $\overline{TU}$ and $\overline{VW}$ are cut by a transversal (say a line that forms angles $\angle U$ and $\angle V$), and since $\angle U=\angle V$, $\overline{TU}\parallel\overline{VW}$.

Answer:

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