Question

tv || qs. complete the proof that ( mangle prs + mangle vuw = 180^{circ} ).

Explanation:

Given \( \overleftrightarrow{TV} \parallel \overleftrightarrow{QS} \), \( \angle VUW \) and \( \angle SRU \) are corresponding angles formed by transversal \( \overleftrightarrow{WP} \) intersecting the parallel lines. By the Corresponding Angles Theorem, corresponding angles are congruent, so \( \angle VUW \cong \angle SRU \). Then, substituting \( m\angle SRU \) with \( m\angle VUW \) in the linear pair sum \( m\angle PRS + m\angle SRU = 180^\circ \) gives the desired result.

Answer:

For Statement 2, the reason is Corresponding Angles Theorem.
For Statement 4, the reason is Substitution Property of Equality (using \( \angle VUW \cong \angle SRU \)).

The completed proof confirms \( m\angle PRS + m\angle VUW = 180^\circ \).