https://file.aiquickdraw.com/mathbot/file/1726493753070z4cqnenz.jpg

Here is how the proof can be filled in, based on the given diagram and problem:

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

**Prove**: \( \overline{XY} \cong \overline{XZ} \)

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**Statements and Reasons**:

1. **Statement**: \( X \) is the midpoint of \( \overline{WY} \)  
   **Reason**: Given.

2. **Statement**: \( WX = XY \)  
   **Reason**: Definition of midpoint (the midpoint divides the segment into two equal parts).

3. **Statement**: \( \overline{WX} \cong \overline{XZ} \)  
   **Reason**: Given.

4. **Statement**: \( WX = XZ \)  
   **Reason**: Definition of congruent segments (congruent segments have equal lengths).

5. **Statement**: \( XY = XZ \)  
   **Reason**: Transitive property of equality (if \( WX = XY \) and \( WX = XZ \), then \( XY = XZ \)).

6. **Statement**: \( \overline{XY} \cong \overline{XZ} \)  
   **Reason**: Definition of congruent segments (segments with equal lengths are congruent).

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Would you like further details on any part of this proof? Here are some related questions for more exploration:

1. How does the transitive property of equality apply to this proof?
2. Why is the definition of a midpoint essential in this proof?
3. Can this proof be used for other geometrical figures, such as triangles?
4. How can this reasoning be extended to three-dimensional geometry?
5. What other properties of congruent segments are useful in geometric proofs?

**Tip**: Always ensure to clearly state given information and define any geometrical properties used in a proof for clarity.

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

Explore a detailed step-by-step proof showing that segment XY is congruent to segment XZ. The problem involves key geometric concepts such as midpoints, congruence, and the transitive property. Perfect for students in Grades 8-10.

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