Question

given: ∠t ≅ ∠v; st || uv prove: tu || vw complete the two - column proof.

Explanation:

Step1: Identify alternate - interior angles

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

Step2: Use the transitive property

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

Step3: Apply the converse of alternate - interior angles theorem

If $\angle U\cong\angle V$, then by the converse of the alternate - interior angles theorem, $\overline{TU}\parallel\overline{VW}$.

Answer:

1. alternate - interior angles theorem; 4. transitive property of congruence; 5. converse of alternate - interior angles theorem