Question

given: (angle t=angle v;overline{st}paralleloverline{uv}) prove: (overline{tu}paralleloverline{vw}) complete the two - column proof.

Explanation:

Step1: Reason for $\angle T=\angle U$

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), and here the transversal is $\overline{TU}$. So the reason for $\angle T = \angle U$ is "Alternate - Interior Angles Theorem".

Step2: Reason for $\angle U=\angle V$

We know that $\angle T=\angle U$ (from step 1) and $\angle T=\angle V$ (given). By the Transitive Property of Equality (if $a = b$ and $a = c$, then $b = c$), we can say $\angle U=\angle V$. So the reason for $\angle U=\angle V$ is "Transitive Property of Equality".

Step3: Reason for $\overline{TU}\parallel\overline{VW}$

Since $\angle U=\angle V$, 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), and here the transversal is $\overline{UV}$. So the reason for $\overline{TU}\parallel\overline{VW}$ is "Converse of the Alternate - Interior Angles Theorem".

Answer:

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