https://file.aiquickdraw.com/mathbot/file/17266774482435b1yzdjc.jpg

Let's complete the proof shown in the image step by step. The goal is to prove that \(\overline{XY} \cong \overline{XZ}\).

### Given:
- \( X \) is the midpoint of \(\overline{WY}\),
- \(\overline{WX} \cong \overline{XZ}\).

### Proof:

| Statements | Reasons |
|------------|----------|
| 1. \( X \) is the midpoint of \(\overline{WY}\) | 1. Given |
| 2. \( WX = XY \) | 2. Definition of midpoint |
| 3. \(\overline{WX} \cong \overline{XZ}\) | 3. Given |
| 4. \( WX = XZ \) | 4. Definition of congruent segments |
| 5. \( XY = XZ \) | 5. Transitive property of equality |
| 6. \(\overline{XY} \cong \overline{XZ}\) | 6. Definition of congruent segments |

This table completes the two-column proof.

Would you like more detailed explanations or have any questions on these steps? Here are some related questions to explore this further:

1. What is the definition of a midpoint in geometry?
2. How does the transitive property of equality work in geometry proofs?
3. Can you explain why \(\overline{XY} \cong \overline{XZ}\) implies equal lengths?
4. What role do congruent segments play in geometric proofs?
5. How would this proof change if \( X \) were not the midpoint?

**Tip:** Always double-check your reasoning and definitions, as they provide the foundation for all proofs!

Prove that XY is congruent to XZ given that X is the midpoint of WY and WX is congruent to XZ.

This problem focuses on proving that XY is congruent to XZ using geometric concepts such as midpoints and congruent segments. It demonstrates key steps like applying the definition of midpoint and congruence, and the transitive property of equality, making it suitable for students in Grades 9-10.

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