To solve this problem, let's analyze the geometry provided in the diagram and the given information.

Given:

• PR and SU are parallel lines.
• $m\angle STQ = 62^\circ$.

We are tasked to find $m\angle RQT$.

Step-by-Step Solution:

1. Identify corresponding angles:

   • Since PR and SU are parallel, and TQ acts as a transversal, the angles formed will have special relationships, such as corresponding angles being equal or alternate interior angles being congruent.

2. Relationship between $\angle STQ$ and $\angle RQT$:

   • $\angle STQ$ and $\angle RQT$ are alternate interior angles because they lie on opposite sides of the transversal TQ, between the parallel lines PR and SU.
   • By the Alternate Interior Angle Theorem, $m\angle STQ = m\angle RQT$.

3. Substitute the given value:

   • $m\angle RQT = m\angle STQ = 62^\circ$.

Final Answer:

$m\angle RQT = 62^\circ.$

If you have further questions or want additional clarification, feel free to ask!

Here are 5 related questions for practice:

1. What is the relationship between corresponding angles in parallel lines?
2. If $m\angle STQ = 45^\circ$, what would $m\angle RQT$ be in this case?
3. What theorem justifies that alternate interior angles are congruent?
4. How do supplementary angles behave along a straight line formed by a transversal?
5. Can the parallel line and transversal properties apply to non-parallel lines?

Tip: Always identify relationships like alternate interior angles, corresponding angles, or vertical angles when working with parallel lines and transversals!