Question

given: ∠t≅∠v; st || uv prove: tu || vw statements reasons 1. st || uv 1 given 2. ∠t≅∠u 2. 3. ∠t≅∠v 3. given 4. ∠u≅∠v 4. 5. tu || vw 5. complete the two - column proof.

Explanation:

Step1: Identify angle - relationship for parallel lines

Since $\overline{ST}\parallel\overline{UV}$, by the alternate - interior angles theorem, $\angle T\cong\angle U$.

Step2: Use the transitive property of congruence

We know that $\angle T\cong\angle V$ (given) and $\angle T\cong\angle U$. By the transitive property of congruence, if $a = b$ and $a = c$, then $b = c$. So, $\angle U\cong\angle V$.

Step3: Prove parallel lines

Since $\angle U\cong\angle V$, by the converse of the alternate - interior angles theorem, if alternate - interior angles are congruent, then the lines are parallel. So, $\overline{TU}\parallel\overline{VW}$.

Answer:

1. alternate - interior angles theorem
2. transitive property of congruence
3. converse of the alternate - interior angles theorem