Question

complete the proof that $\triangle vwy \cong \triangle uxy$.
1 $\overline{xy} \cong \overline{wy}$ reason: given
2 $\overline{vy} \cong \overline{uy}$ reason: given
3 $\angle uyx \cong \angle vyw$ reason:
4 $\triangle vwy \cong \triangle uxy$ reason:

Explanation:

Step1: Analyze ∠UYX and ∠VYW

∠UYX and ∠VYW are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason for ∠UYX ≅ ∠VYW is "Vertical Angles are Congruent".

Step2: Determine Triangle Congruence

We have two sides and the included angle: $\overline{XY} \cong \overline{WY}$, $\overline{VY} \cong \overline{UY}$, and the included angle ∠UYX ≅ ∠VYW. By the Side - Angle - Side (SAS) Congruence Postulate, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. So the reason for $\triangle VWY \cong \triangle UXY$ is "SAS Congruence Postulate".

Answer:

For row 3, Reason: Vertical Angles are Congruent
For row 4, Reason: SAS Congruence Postulate