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To complete the two-column proof for the given problem, let's analyze each statement and reason accordingly.

### Given Information:
1. \( WZ \) bisects \( \angle WXY \).
2. \( WY \cong WX \).

The objective is to prove that \( \triangle WXZ \cong \triangle WYZ \).

### Proof Structure
Using the statements provided, let's identify the reasons for each one:

1. **Statement:** \( WZ \) bisects \( \angle WXY \)  
   **Reason:** Given (1)  

2. **Statement:** \( WY \cong WX \)  
   **Reason:** Given (2)  

3. **Statement:** \( \angle XWZ \cong \angle YWZ \)  
   **Reason:** Definition of Angle Bisector  

4. **Statement:** \( WZ \cong WZ \)  
   **Reason:** Reflexive Property of Congruence  

5. **Statement:** \( \triangle WXZ \cong \triangle WYZ \)  
   **Reason:** SAS (Side-Angle-Side)

### Completed Proof Table

| Statement                     | Reason                             |
|-------------------------------|------------------------------------|
| 1. \( WZ \) bisects \( \angle WXY \) | Given (1)                         |
| 2. \( WY \cong WX \)                 | Given (2)                         |
| 3. \( \angle XWZ \cong \angle YWZ \) | Definition of Angle Bisector      |
| 4. \( WZ \cong WZ \)                 | Reflexive Property of Congruence  |
| 5. \( \triangle WXZ \cong \triangle WYZ \) | SAS                         |

This matches each statement with the appropriate reason, completing the two-column proof.

Would you like more details or have any questions about this proof?

Here are some related questions that can expand your understanding:
1. Why is the Reflexive Property of Congruence used in this proof?
2. How does the definition of an angle bisector apply to prove congruence?
3. What does the SAS (Side-Angle-Side) Congruence Postulate imply in triangle proofs?
4. Can there be alternate methods to prove triangle congruence in this scenario?
5. Why are given statements essential in formal proofs?

**Tip:** Always start with given information when organizing a proof, as it provides the foundation for subsequent reasoning.

Complete the two-column proof where WZ bisects angle WXY and WY is congruent to WX. Prove that triangle WXZ is congruent to triangle WYZ.

Explore a detailed two-column proof to show triangle congruence using an angle bisector and SAS postulate. Given WZ bisects angle WXY and WY is congruent to WX, complete the proof that triangle WXZ is congruent to triangle WYZ. This solution applies the Definition of Angle Bisector, Reflexive Property, and SAS Congruence Postulate, ideal for high school geometry students.

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