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Let's analyze this proof problem and complete the proof to show that \( \overline{WY} \cong \overline{UY} \).

### Given Information:
1. \( UX \perp \overline{UW} \)
2. \( UX = VW \)
3. \( UW \perp VW \)
4. \( \angle U \cong \angle W \) (All right angles are congruent)

### Completing the Proof:
For the statements and reasons:

**Step 5: \( \angle UYX \cong \angle VYW \)**  
**Reason:** Vertical angles are congruent.

**Step 6: \( \triangle UYX \cong \triangle VWY \)**  
**Reason:** AAS (Angle-Angle-Side) Congruence Postulate. This is because two angles (\( \angle U \) and \( \angle UYX \)) and the side \( UX \) are congruent to their counterparts in the other triangle (\( \angle W \), \( \angle VYW \), and \( VW \)).

**Step 7: \( \overline{WY} \cong \overline{UY} \)**  
**Reason:** Corresponding parts of congruent triangles (CPCTC).

### Summary of the Completed Proof:
1. \( UX \perp UW \) — Given
2. \( UX = VW \) — Given
3. \( UW \perp VW \) — Given
4. \( \angle U \cong \angle W \) — All right angles are congruent
5. \( \angle UYX \cong \angle VYW \) — Vertical angles are congruent
6. \( \triangle UYX \cong \triangle VWY \) — AAS Congruence Postulate
7. \( WY \cong UY \) — CPCTC

Would you like more details on any of these steps or concepts?

### Related Questions:
1. What is the AAS (Angle-Angle-Side) Congruence Postulate?
2. Why are all right angles considered congruent?
3. How do vertical angles relate to congruence in geometry proofs?
4. What is CPCTC and how is it applied in triangle congruence?
5. How can we prove triangle congruence using other methods like SAS or SSS?

### Tip:
Remember, when proving triangle congruence, identifying pairs of congruent angles or sides can simplify the proof. Look for given conditions like perpendicularity or parallel lines to find congruent angles.

Complete the proof that \( WY \cong UY \), using the given statements and reasons in the geometry problem.

This problem demonstrates how to prove that segment WY is congruent to segment UY using triangle congruence principles such as AAS (Angle-Angle-Side) and the CPCTC theorem. It includes key geometry concepts and is suitable for students in Grades 9-10.

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