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We are given a diagram with parallel lines \( PR \parallel SU \) and an angle \( m\angle UTQ = 63^\circ \). The task is to find the measure of \( m\angle PQT \).

### Step-by-step solution:

1. **Identify the relevant angles:**
   - The lines \( PR \parallel SU \) are cut by the transversal line \( TQ \), which creates alternate interior angles.

2. **Relation between \( \angle UTQ \) and \( \angle PQT \):**
   - Since \( PR \parallel SU \) and \( TQ \) is a transversal, the angle \( \angle UTQ \) is alternate interior with \( \angle PQT \).
   - Alternate interior angles are congruent when lines are parallel.

3. **Conclusion:**
   - \( m\angle PQT = m\angle UTQ = 63^\circ \).

Thus, the measure of \( m\angle PQT \) is \( \boxed{63^\circ} \).

Would you like more details or have any questions? Here are some related questions for practice:

1. What is the definition of alternate interior angles?
2. How do we determine if two lines are parallel in a geometric diagram?
3. If \( m\angle UTQ \) was given as \( 45^\circ \), what would \( m\angle PQT \) be?
4. What role do transversals play when working with parallel lines?
5. How can you use corresponding angles to solve for unknown angles in a parallel line setup?

**Tip:** In problems involving parallel lines and transversals, always look for congruent or supplementary angle pairs like alternate interior or corresponding angles.

If PR and SU are parallel lines and m∠UTQ = 63°, what is m∠PQT?

Find the measure of ∠PQT using the Alternate Interior Angles Theorem, given parallel lines PR and SU and m∠UTQ = 63°. This problem is ideal for students in Grades 7-9 and covers key geometric concepts related to angles and transversals.

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