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In the given problem, the goal is to complete the proof that \(\angle RUT \cong \angle SRU\).

The steps in the proof are nearly completed. Let's look at what is provided and what the missing reason might be:

### Step-by-step analysis:
1. **\( \overline{TV} \parallel \overline{QS} \)** is given as a parallel condition.
   
2. **\(\angle RUT \cong \angle PRQ\)** is established by the **Corresponding Angles Theorem**, which applies because \( \overline{TV} \parallel \overline{QS} \).

3. **\(\angle PRQ \cong \angle SRU\)** is concluded by the **Vertical Angle Theorem** because \(\overline{PR}\) and \(\overline{QS}\) cross each other at point \(R\).

4. The last step is **\(\angle RUT \cong \angle SRU\)**, and the reason for this is the **Transitive Property of Congruence**.

### Missing reason:
- **Transitive Property of Congruence**: If \(\angle RUT \cong \angle PRQ\) and \(\angle PRQ \cong \angle SRU\), then by transitivity, \(\angle RUT \cong \angle SRU\).

### Completed proof:

| Statement                           | Reason                         |
|-------------------------------------|--------------------------------|
| 1. \( \overline{TV} \parallel \overline{QS} \)  | Given                           |
| 2. \( \angle RUT \cong \angle PRQ \)           | Corresponding Angles Theorem    |
| 3. \( \angle PRQ \cong \angle SRU \)           | Vertical Angle Theorem          |
| 4. \( \angle RUT \cong \angle SRU \)           | Transitive Property of Congruence |

Would you like further details or have any questions?

### Related Questions:
1. What is the Corresponding Angles Theorem, and when is it used?
2. Can you explain the Transitive Property of Congruence in geometry?
3. What are vertical angles, and how are they proven congruent?
4. What are the conditions for applying the Corresponding Angles Theorem?
5. How do parallel lines influence angle relationships in geometry?

### Tip:
When working with angle proofs, look for patterns like corresponding angles and vertical angles, as they often provide quick ways to establish congruence in geometric diagrams.

Complete the proof that ∠RUT ≅ ∠SRU. Given that TV ∥ QS and angles are formed by the intersections of lines.

Learn how to complete the proof that ∠RUT is congruent to ∠SRU using the Corresponding Angles Theorem, Vertical Angle Theorem, and the Transitive Property of Congruence. This geometry problem involves parallel lines and congruent angles, making it suitable for students in Grades 9-12.

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