Question

complete the proof that ∠vuw≅∠prq.
statement reason
1 qs || tv given
2 ∠vuw≅∠sru
3 ∠sru≅∠prq
4 ∠vuw≅∠prq

Explanation:

Step1: Identify angle - pair relationship for step 2

When two parallel lines ($\overleftrightarrow{QS}\parallel\overleftrightarrow{TV}$) are cut by a transversal, alternate - interior angles are congruent. $\angle VUW$ and $\angle SRU$ are alternate - interior angles.
So the reason for $\angle VUW\cong\angle SRU$ is "Alternate interior angles are congruent when two parallel lines are cut by a transversal".

Step2: Identify angle - pair relationship for step 3

$\angle SRU$ and $\angle PRQ$ are vertical angles. Vertical angles are always congruent.
So the reason for $\angle SRU\cong\angle PRQ$ is "Vertical angles are congruent".

Step3: Use the transitive property for step 4

Since $\angle VUW\cong\angle SRU$ and $\angle SRU\cong\angle PRQ$, by the transitive property of congruence (if $a = b$ and $b = c$, then $a = c$ for angles, if $\angle A\cong\angle B$ and $\angle B\cong\angle C$, then $\angle A\cong\angle C$), we have $\angle VUW\cong\angle PRQ$.
So the reason for $\angle VUW\cong\angle PRQ$ is "Transitive property of congruence".

Answer:

1. Alternate interior angles are congruent when two parallel lines are cut by a transversal
2. Vertical angles are congruent
3. Transitive property of congruence