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To complete the proof that \( \angle PRQ \cong \angle VUW \), let's examine the steps and reasons required.

### Proof Table Completion:

1. **Given:** \( QS \parallel TV \).
2. **Statement:** \( \angle VUW \cong \angle SRU \)
   - **Reason:** **Alternate Interior Angles Theorem**. Since \( QS \parallel TV \) and \( UW \) is a transversal, \( \angle VUW \) and \( \angle SRU \) are alternate interior angles and therefore congruent.
3. **Statement:** \( \angle SRU \cong \angle PRQ \)
   - **Reason:** **Vertical Angles Theorem**. \( \angle SRU \) and \( \angle PRQ \) are vertical angles (formed by intersecting lines \( SR \) and \( PR \)), and vertical angles are always congruent.
4. **Statement:** \( \angle PRQ \cong \angle VUW \)
   - **Reason:** **Transitive Property of Congruence**. Since \( \angle VUW \cong \angle SRU \) and \( \angle SRU \cong \angle PRQ \), it follows by the transitive property that \( \angle PRQ \cong \angle VUW \).

The completed proof table should look like this:

| Statement                     | Reason                          |
|-------------------------------|---------------------------------|
| 1. \( QS \parallel TV \)      | Given                           |
| 2. \( \angle VUW \cong \angle SRU \) | Alternate Interior Angles Theorem |
| 3. \( \angle SRU \cong \angle PRQ \) | Vertical Angles Theorem       |
| 4. \( \angle PRQ \cong \angle VUW \) | Transitive Property of Congruence |

Would you like more details on any of the theorems used, or have any questions?

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### Follow-up Questions:
1. What is the Alternate Interior Angles Theorem, and when is it used?
2. How does the Vertical Angles Theorem apply in geometry?
3. What is the Transitive Property of Congruence, and how does it simplify proofs?
4. How do we identify transversals in geometry problems involving parallel lines?
5. Can we apply these properties to non-parallel line configurations?

### Tip:
When proving angle congruence, always check for alternate interior angles, corresponding angles, and vertical angles in diagrams involving parallel lines and transversals.

Complete the proof that ∠PRQ ≅ ∠VUW.

Learn how to complete the proof that ∠PRQ ≅ ∠VUW with a step-by-step solution using the Alternate Interior Angles Theorem, Vertical Angles Theorem, and Transitive Property of Congruence. This problem is suitable for students in Grades 9-10.

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