Question

qs || xz and tv || xz. complete the proof that ∠vuw ≅ ∠prq.
statement\treason
1 qs || xz\tgiven
2 tv || xz\tgiven
3 ∠vuw ≅ ∠wyz\tcorresponding angles theorem
4 ∠wyz ≅ ∠prq
5 ∠vuw ≅ ∠prq\talternate exterior angles theorem
\talternate interior angles theorem
\tconverse of alternate exterior angles theorem
\tconverse of alternate interior angles theorem
\tconverse of corresponding angles theorem
\tconverse of same - side exterior angles theorem
\tconverse of same - side interior angles theorem
\tcorresponding angles theorem
\tsame - side exterior angles theorem
\tsame - side interior angles theorem

Explanation:

Step1: Identify parallel - line relationships

Given $\overleftrightarrow{QS}\parallel\overleftrightarrow{XZ}$ and $\overleftrightarrow{TV}\parallel\overleftrightarrow{XZ}$.

Step2: Use corresponding - angles theorem

Since $\overleftrightarrow{TV}\parallel\overleftrightarrow{XZ}$, by the Corresponding Angles Theorem, $\angle VUW\cong\angle WYZ$.

Step3: Use corresponding - angles theorem again

Since $\overleftrightarrow{QS}\parallel\overleftrightarrow{XZ}$, $\angle WYZ\cong\angle PRQ$ by the Corresponding Angles Theorem.

Step4: Apply transitive property of congruence

If $\angle VUW\cong\angle WYZ$ and $\angle WYZ\cong\angle PRQ$, then $\angle VUW\cong\angle PRQ$ (transitive property of congruence).

Answer:

The reason for statement 4 is "Corresponding Angles Theorem".