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The proof you are working on is aimed at demonstrating that triangles \( \triangle VYZ \) and \( \triangle XWZ \) are congruent. Based on the information provided, here is how the proof can be completed step by step.

### Complete Proof Steps:
1. **Statement:** \( Z \) is the midpoint of \( VX \).
   - **Reason:** Given.
   
2. **Statement:** \( Z \) is the midpoint of \( WY \).
   - **Reason:** Given.

3. **Statement:** \( VZ \cong XZ \).
   - **Reason:** Definition of midpoint (since \( Z \) is the midpoint of \( VX \), the segments \( VZ \) and \( XZ \) are congruent).

4. **Statement:** \( WZ \cong YZ \).
   - **Reason:** Definition of midpoint (since \( Z \) is the midpoint of \( WY \), the segments \( WZ \) and \( YZ \) are congruent).

5. **Statement:** \( \angle VZY \cong \angle WZX \).
   - **Reason:** Vertical Angle Theorem (the angles are vertical angles as the two lines cross at point \( Z \)).

6. **Statement:** \( \triangle VYZ \cong \triangle XWZ \).
   - **Reason:** Side-Angle-Side (SAS) Congruence Theorem (using the congruent sides \( VZ \cong XZ \), \( WZ \cong YZ \), and the congruent angles \( \angle VZY \cong \angle WZX \)).

This completes the proof that \( \triangle VYZ \cong \triangle XWZ \).

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Would you like more details or explanations on any step? Here are five related questions to expand on this:

1. What is the Side-Angle-Side (SAS) Congruence Theorem?
2. How does the Vertical Angle Theorem work in triangle congruence proofs?
3. What is the significance of midpoints in proving triangle congruence?
4. Could other triangle congruence postulates (like SSS or ASA) be applied to this problem?
5. How does the congruence of these two triangles help in solving other geometric problems?

**Tip:** In many proofs, focusing on congruent angles and corresponding sides can often lead to a quicker resolution when using congruence theorems such as SAS or SSS.

Complete the proof that triangle VYZ is congruent to triangle XWZ given that Z is the midpoint of VX and WY.

This page provides a step-by-step geometric proof that triangle VYZ is congruent to triangle XWZ by using the midpoint of two lines and the Vertical Angle Theorem. This problem is suitable for students in Grades 8-10 learning about triangle congruence and basic geometric theorems.

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