https://file.aiquickdraw.com/mathbot/file/1726749061254d3ssjrij.jpg

The goal is to complete the two-column proof to show that \( XY \cong ZW \). Here's the detailed breakdown based on the given information and the image:

### Given:
- Point \( Y \) is the midpoint of \( XZ \).
- Point \( W \) is collinear with \( X \), \( Y \), and \( Z \).
- \( Z \) is the midpoint of \( YW \).

We need to prove: \( XY \cong ZW \).

### Filling in the two-column proof:

| **Statements**                                   | **Reasons**                         |
|--------------------------------------------------|-------------------------------------|
| 1. \( Y \) is the midpoint of \( XZ \) and \( Z \) is the midpoint of \( YW \). | 1. Given                              |
| 2. \( XY \cong YZ \) and \( YZ \cong ZW \)       | 2. Midpoint Theorem                 |
| 3. \( XY \cong ZW \)                             | 3. Definition of congruent segments |
| 4. \( XY \cong ZW \)                             | 4. Transitive Property of Equality  |
| 5. \( XY \cong ZW \)                             | 5. Definition of congruent segments |

### Explanation:
1. The first statement reiterates the given information about the midpoints of segments \( XZ \) and \( YW \).
2. By the midpoint theorem, the midpoint divides a segment into two congruent parts, so \( XY \cong YZ \) and \( YZ \cong ZW \).
3. This step defines that if two segments are congruent, the segments themselves are congruent as well.
4. Using the Transitive Property of Equality, if \( XY \cong YZ \) and \( YZ \cong ZW \), then \( XY \cong ZW \).

Let me know if you need further details or clarifications.

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Here are five related questions to deepen understanding:
1. How does the midpoint theorem work in geometry?
2. What is the transitive property in the context of geometric proofs?
3. What other properties or theorems could help prove segment congruence?
4. How can this type of proof be applied in other geometric configurations?
5. What is the importance of clearly stating reasons in a two-column proof?

**Tip**: Always clearly define your given information before starting a proof. This helps structure the logical flow.

PROOF: Point Y is the midpoint of XZ. Point W is collinear with X, Y, and Z. Z is the midpoint of YW. Complete the two-column proof to prove that XY ≅ ZW.

Solve the geometric proof where Y is the midpoint of XZ and Z is the midpoint of YW. Learn how to apply the midpoint theorem and transitive property to prove that XY ≅ ZW. Suitable for students in grades 9-12.

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