Question

∠rut ≅ ∠sru. complete the proof that (overleftrightarrow{tv}paralleloverleftrightarrow{qs}).
statement reason
1 ∠rut ≅ ∠sru given
2 ∠prq ≅ ∠sru vertical angle theorem
3 ∠rut ≅ ∠prq transitive property of congruence
4 (overleftrightarrow{tv}paralleloverleftrightarrow{qs}) options: angles forming a linear pair sum to 180°, converse of corresponding angles theorem, corresponding angles theorem, definition of supplementary angles, vertical angle theorem

Explanation:

Step1: Identify given and proven congruent angles

We are given $\angle{RUT}\cong\angle{SRU}$ and have proven $\angle{RUT}\cong\angle{PRQ}$ through the vertical - angle theorem and transitive property. $\angle{PRQ}$ and $\angle{RUT}$ are corresponding angles with respect to lines $\overleftrightarrow{TV}$ and $\overleftrightarrow{QS}$ and transversal $\overleftrightarrow{WP}$.

Step2: Apply angle - congruence and parallel - line relationship

If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. Since $\angle{RUT}\cong\angle{PRQ}$, by the Converse of Corresponding Angles Theorem, $\overleftrightarrow{TV}\parallel\overleftrightarrow{QS}$.

Answer:

Converse of Corresponding Angles Theorem