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The problem is to complete the proof that triangles \( \Delta WXZ \cong \Delta YVZ \). We are given two parallel line segments, \( WX \parallel VY \), and that \( VY \cong WX \).

Here is how the proof can be completed:

### Statements and Reasons:

1. \( VY \cong WX \) &nbsp;&nbsp;&nbsp;**(Given)**
2. \( WX \parallel VY \) &nbsp;&nbsp;&nbsp;**(Given)**
3. \( \angle VZY \cong \angle WZX \) &nbsp;&nbsp;&nbsp;**(Alternate Interior Angles, since \( WX \parallel VY \))**
4. \( \angle Y \cong \angle W \) &nbsp;&nbsp;&nbsp;**(Vertical Angles are congruent)**
5. \( \Delta WXZ \cong \Delta YVZ \) &nbsp;&nbsp;&nbsp;**(ASA: Angle-Side-Angle postulate)**

### Explanation:
- The two triangles share a pair of alternate interior angles because the sides \( WX \parallel VY \) and \( Z \) is a transversal.
- The vertical angles at \( Z \) are congruent.
- Since \( WX \cong VY \) (given) and the two pairs of angles are congruent, we use the Angle-Side-Angle (ASA) postulate to prove that the two triangles are congruent.

Would you like more detailed explanations, or do you have any questions? 

Here are some related questions to explore:
1. How do alternate interior angles prove congruence?
2. What other congruence postulates could be applied if different information were given?
3. How does the ASA postulate differ from the SAS postulate?
4. Why are vertical angles always congruent?
5. Can you apply this proof method to non-parallel lines?

**Tip**: When working with parallel lines and transversals, always look for alternate interior angles as a potential clue for congruence.

Complete the proof that ΔWXZ ≅ ΔYVZ, given WX ∥ VY and VY ≅ WX.

Learn how to prove that triangles ΔWXZ and ΔYVZ are congruent using the ASA postulate. This proof involves parallel lines, alternate interior angles, and vertical angles, making it suitable for students in Grades 8-10.

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