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To solve for \( m\angle RQO \) in the given diagram, we need to analyze the geometry step by step:

### Key Information:
- \( \overline{PR} \parallel \overline{SU} \) (parallel lines)
- \( m\angle UTV = 57.9^\circ \)
- We are looking for \( m\angle RQO \).

### Explanation:

1. Since \( \overline{PR} \parallel \overline{SU} \) and \( \overline{TQ} \) is a transversal, angles \( \angle UTV \) and \( \angle RQO \) are **alternate interior angles**.
  
2. By the **Alternate Interior Angles Theorem**, if two parallel lines are cut by a transversal, the alternate interior angles are equal. Therefore:
   \[
   m\angle RQO = m\angle UTV = 57.9^\circ
   \]

Thus, \( m\angle RQO = 57.9^\circ \).

Let me know if you'd like more details or have any questions!

Here are some related questions:
1. What is the Alternate Interior Angles Theorem?
2. How do you identify a transversal in a diagram?
3. Can you provide an example where alternate interior angles are not congruent?
4. How does the Parallel Postulate apply in this diagram?
5. What would change if the lines were not parallel?

**Tip:** When dealing with parallel lines and a transversal, always look for corresponding and alternate interior angles for quick solutions!

If PR and SU are parallel lines and m∠UTV = 57.9°, what is m∠RQO?

This problem demonstrates how to calculate the measure of angle RQO by applying the Alternate Interior Angles Theorem. Given parallel lines and a transversal, we can determine that m∠RQO = 57.9°. Ideal for students in Grades 8-10 studying geometry and angle relationships.

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