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!

Given: X is the midpoint of segment WY, and WX is congruent to XZ. Prove: XY is congruent to XZ.

Explore how to prove segment congruence by using midpoints and congruent segments in geometry. This proof demonstrates key geometric principles such as the definition of midpoints and the transitive property of equality. Ideal for students in Grades 7-9 learning about basic geometric proofs.

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