In the diagram provided, we are told that $\overline{OQ} \parallel \overline{RT}$, and the angle $\angle TSU = 70^\circ$. The question asks us to find the measure of $\angle QPS$.

Analyzing the situation:

• $\overline{OQ}$ and $\overline{RT}$ are parallel lines.
• $\overline{PN}$ serves as a transversal to these parallel lines.
• $\angle TSU$ is provided as $70^\circ$.

Key Concepts:

• Alternate Interior Angles: When a transversal crosses two parallel lines, alternate interior angles are equal.
• Corresponding Angles: Angles in the same relative position at each intersection where a transversal crosses parallel lines are congruent.

From the diagram, $\angle TSU$ corresponds to $\angle QPS$, as both are alternate interior angles formed by the transversal $\overline{PS}$ cutting through the parallel lines $\overline{OQ}$ and $\overline{RT}$.

Conclusion:

Since $\overline{OQ} \parallel \overline{RT}$ and $\overline{PN}$ is a transversal, we can conclude that: $\angle QPS = \angle TSU = 70^\circ$

Thus, the measure of $\angle QPS$ is $70^\circ$.

Do you need any more details or clarifications on this?

Here are 5 related questions to further your understanding:

1. What are corresponding angles, and how can they be identified?
2. How do alternate interior angles relate to parallel lines?
3. Can you explain why supplementary angles are relevant in transversal problems?
4. What is the role of vertical angles in solving problems with parallel lines and transversals?
5. How can you determine angles in polygons using parallel line properties?

Tip: Always label angles in diagrams carefully to visualize relationships between them when dealing with parallel lines and transversals.