Question

statement\treason
1 (overleftrightarrow{eg}paralleloverleftrightarrow{vx})\tgiven
2 (overleftrightarrow{eg}paralleloverleftrightarrow{su})\tgiven
3 (angle xwycongangle gfy)\tcorresponding angles theorem
4 (angle gfycongangle rts)\t
5 (angle rtscongangle xwy)\ttransitive property of congruence

Explanation:

Step1: Identify parallel - line relationships

We are given $\overleftrightarrow{EG}\parallel\overleftrightarrow{VX}$ and $\overleftrightarrow{EG}\parallel\overleftrightarrow{SU}$ as initial facts.

Step2: Apply corresponding - angles theorem

Since $\overleftrightarrow{EG}\parallel\overleftrightarrow{VX}$, by the Corresponding Angles Theorem, $\angle XWY\cong\angle GFY$.

Step3: Find another pair of congruent angles

Since $\overleftrightarrow{EG}\parallel\overleftrightarrow{SU}$, and considering the relevant transversal, $\angle GFY\cong\angle RTS$ by the Corresponding Angles Theorem.

Step4: Use transitive property

We know that if $\angle XWY\cong\angle GFY$ and $\angle GFY\cong\angle RTS$, then by the Transitive Property of Congruence, $\angle RTS\cong\angle XWY$.

Answer:

The reasoning for each step is correct. The proof shows that $\angle RTS\cong\angle XWY$ using the given parallel - line relationships, the Corresponding Angles Theorem, and the Transitive Property of Congruence.