Question

qs || tv. complete the proof that ∠prs ≅ ∠tuw.
statement reason
1 qs || tv given
2 ∠tuw ≅ ∠qru corresponding angles theorem
3 ∠qru ≅ ∠prs
4 ∠prs ≅ ∠tuw

Explanation:

Step1: Identify vertical - angle property

$\angle QRU$ and $\angle PRS$ are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So, $\angle QRU\cong\angle PRS$.

Step2: Use the transitive property

Since $\angle TUW\cong\angle QRU$ (from Step 2 in the given table) and $\angle QRU\cong\angle PRS$ (from Step 1 above), by the Transitive Property of Congruence, if $\angle A\cong\angle B$ and $\angle B\cong\angle C$, then $\angle A\cong\angle C$. Here, $A = \angle TUW$, $B=\angle QRU$, and $C = \angle PRS$. So, $\angle PRS\cong\angle TUW$.

Answer:

1. Reason: Vertical Angles Theorem
2. Reason: Transitive Property of Congruence