Question

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statement reason
1 (overleftrightarrow{tv}paralleloverleftrightarrow{xz}) given
2 (overleftrightarrow{xz}paralleloverleftrightarrow{qs}) given
3 (angle tuwcongangle wyx)
4 (angle wyxcongangle prs) alternate interior angles theorem
5 (angle tuwcongangle prs) alternate exterior angles theorem
converse of alternate exterior angles theorem
converse of alternate interior angles theorem
converse of corresponding angles theorem
converse of same - side exterior angles theorem
converse of same - side interior angles theorem
corresponding angles theorem
same - side exterior angles theorem
same - side interior angles theorem

Explanation:

Step1: Identify parallel - line relationships

We know $\overleftrightarrow{TV}\parallel\overleftrightarrow{XZ}$ and $\overleftrightarrow{XZ}\parallel\overleftrightarrow{QS}$.

Step2: Analyze $\angle TUW$ and $\angle WYX$

$\angle TUW$ and $\angle WYX$ are corresponding angles formed by parallel lines $\overleftrightarrow{TV}$ and $\overleftrightarrow{XZ}$ with transversal $\overleftrightarrow{PW}$. By the Corresponding Angles Theorem, when two parallel lines are cut by a transversal, corresponding angles are congruent. So the reason for $\angle TUW\cong\angle WYX$ is the Corresponding Angles Theorem.

Answer:

Corresponding Angles Theorem