Question

$$ \overline { v y } \perp \overline { w x } $$ and $$ \angle x v y \cong \angle w v y $$. complete the proof that $$ \overline { x y } \cong \overline { w y } $$.

statement	reason
2 $$ \angle x v y \cong \angle w v y $$	given
3 $$ \angle v y w \cong \angle v y x $$	all right angles are congruent
4 $$ \overline { v y } \cong \overline { v y } $$	reflexive property of congruence
5

Explanation:

Step1: Identify congruent triangles

We have two triangles: $\triangle VYW$ and $\triangle VYX$. From the given: $\angle XVY \cong \angle WVY$ (statement 2), $\overline{VY} \cong \overline{VY}$ (reflexive property, statement 4), and $\angle VYW \cong \angle VYX$ (right angles, statement 3). These satisfy ASA congruence.

Step2: Conclude corresponding sides congruent

By ASA, $\triangle VYW \cong \triangle VYX$. Corresponding sides $\overline{XY}$ and $\overline{WY}$ are congruent.

Answer:

$\overline{XY} \cong \overline{WY}$ (by ASA congruence of $\triangle VYW$ and $\triangle VYX$)