https://file.aiquickdraw.com/mathbot/file/1728341332493squiu5vy.jpg

It looks like you need help completing the proof to show that \(\angle WXZ \cong \angle VYZ\). Let’s break this down step-by-step based on the given diagram.

### Proof of \(\angle WXZ \cong \angle VYZ\):
This type of proof typically involves parallel lines and corresponding angles. From the diagram, we can see that two lines are intersected by a transversal.

1. **Statement:** \(\angle XWZ \cong \angle YVZ\) 
   - **Reason:** Given.

2. **Statement:** \(WX \parallel VY\)
   - **Reason:** This is likely a given property in the diagram, suggesting that the lines \(WX\) and \(VY\) are parallel.

3. **Statement:** \(\angle YVZ \cong \angle WXZ\)
   - **Reason:** Corresponding Angles Postulate. Since \(WX \parallel VY\), corresponding angles formed by the transversal \(XZ\) and \(YZ\) are congruent.

4. **Statement:** \(\angle XWZ \cong \angle VYZ\)
   - **Reason:** Vertical Angles Theorem. In the diagram, \(\angle XWZ\) and \(\angle VYZ\) are vertical angles and vertical angles are always congruent.

5. **Statement:** \(\angle YVZ \cong \angle VYZ\)
   - **Reason:** Transitive Property. From earlier statements, we can now conclude that these angles are congruent by the transitive property.

6. **Statement:** \(\angle WXZ \cong \angle VYZ\)
   - **Reason:** Definition of congruent angles. We've now proven that these two angles are congruent, completing the proof.

### Final Answer:
Here is the complete proof in the table format based on your image:

| Statement                           | Reason                                |
|--------------------------------------|---------------------------------------|
| 1. \(\angle XWZ \cong \angle YVZ\)   | Given                                 |
| 2. \(WX \parallel VY\)               | Given                                 |
| 3. \(\angle YVZ \cong \angle WXZ\)   | Corresponding Angles Postulate         |
| 4. \(\angle XWZ \cong \angle VYZ\)   | Vertical Angles Theorem               |
| 5. \(\angle YVZ \cong \angle VYZ\)   | Transitive Property of Congruence      |
| 6. \(\angle WXZ \cong \angle VYZ\)   | Definition of congruent angles         |

Would you like further clarification or have any questions?

### Related Questions:
1. What is the Corresponding Angles Postulate?
2. How does the Transitive Property of congruence work in proofs?
3. Why are vertical angles always congruent?
4. How do we prove parallel lines using angle relationships?
5. What other theorems are related to transversal lines and parallel lines?

### Tip:
When proving angles congruent in diagrams with parallel lines, look for transversal lines. Corresponding, alternate interior, or alternate exterior angles formed by a transversal with parallel lines are congruent by specific postulates and theorems!

Complete the proof that ∠WXZ ≅ ∠VYZ.

Complete the geometry proof that ∠WXZ ≅ ∠VYZ by applying key theorems such as the Corresponding Angles Postulate and the Vertical Angles Theorem. Learn how transversals and parallel lines create congruent angles in this step-by-step guide, suitable for students in Grades 8-10.

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