Question

statement reason 1. $overleftrightarrow{vx}paralleloverleftrightarrow{ik}$ given 2. $overleftrightarrow{ik}paralleloverleftrightarrow{su}$ given 3. $angle rwvcongangle ijr$ alternate exterior angles theorem 4. $angle ijrcongangle uty$ alternate interior angles theorem 5. $angle rwvcongangle uty$ converse of alternate exterior angles theorem converse of alternate interior angles theorem converse of corresponding angles theorem

Explanation:

Step1: Identify parallel lines

We are given that $\overleftrightarrow{VX}\parallel\overleftrightarrow{IK}$ and $\overleftrightarrow{IK}\parallel\overleftrightarrow{SU}$.

Step2: Analyze angle - angle relationships

$\angle RWV$ and $\angle IJR$ are alternate exterior angles with respect to the parallel lines $\overleftrightarrow{VX}$ and $\overleftrightarrow{IK}$ cut by the transversal $\overleftrightarrow{RY}$, so by the Alternate Exterior Angles Theorem, $\angle RWV\cong\angle IJR$. $\angle IJR$ and $\angle UTY$ are alternate interior angles with respect to the parallel lines $\overleftrightarrow{IK}$ and $\overleftrightarrow{SU}$ cut by the transversal $\overleftrightarrow{RY}$, so by the Alternate Interior Angles Theorem, $\angle IJR\cong\angle UTY$.

Step3: Use the transitive property of congruence

Since $\angle RWV\cong\angle IJR$ and $\angle IJR\cong\angle UTY$, by the transitive property of congruence, $\angle RWV\cong\angle UTY$.

Answer:

The reason for $\angle RWV\cong\angle UTY$ is the Transitive Property of Congruence (since we first used the Alternate Exterior Angles Theorem to get $\angle RWV\cong\angle IJR$ and the Alternate Interior Angles Theorem to get $\angle IJR\cong\angle UTY$).