Let's work through the problem. Based on the image, you are tasked with completing the proof that $\overline{VX} \cong \overline{VW}$ using the statements and reasons provided.

Steps for Completing the Proof:

1. Statement: $\overline{VY} \perp \overline{WX}$
   Reason: Given. (The diagram shows $\overline{VY}$ is perpendicular to $\overline{WX}$.)

2. Statement: $\angle LW \cong \angle LX$
   Reason: Isosceles triangles have congruent base angles. ($\triangle LWX$ is isosceles.)

3. Statement: $\angle LYW \cong \angle LYX$
   Reason: Corresponding angles of congruent triangles are equal. ($\triangle LYW \cong \triangle LYX$ by the Angle-Angle Similarity postulate.)

4. Statement: $\overline{VY} \cong \overline{VY}$
   Reason: Reflexive property of congruence. (Any segment is congruent to itself.)

5. Statement: $\triangle LWY \cong \triangle LXY$
   Reason: ASA (Angle-Side-Angle) Congruence Postulate. (Two angles and the included side are congruent.)

6. Statement: $\overline{VX} \cong \overline{VW}$
   Reason: Corresponding parts of congruent triangles (CPCTC).

Would you like me to clarify any of these steps in more detail?

Related Questions:

1. What are other methods to prove triangle congruence besides ASA?
2. How does the CPCTC principle work in proving segment congruence?
3. Why is the reflexive property important in geometric proofs?
4. Can perpendicular segments help establish triangle similarity or congruence?
5. What role do isosceles triangles play in geometric proofs?

Tip:

Always check if congruent triangles share a common side or angle to simplify proofs using the reflexive property.