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The diagram presents two parallel lines \( OQ \) and \( RT \), intersected by a transversal \( UP \), and an angle \( \angle RSP = 110^\circ \) is given. The task is to find the measure of \( \angle OPS \).

Since \( OQ \parallel RT \) and \( UP \) is the transversal, we can use properties of alternate interior angles and linear pairs:

1. \( \angle RSP \) and \( \angle OPS \) are consecutive interior angles (same-side interior angles) along the transversal \( UP \), meaning their sum must be \( 180^\circ \) (since the lines are parallel).
   
Thus:
\[
\angle RSP + \angle OPS = 180^\circ
\]
Substitute \( \angle RSP = 110^\circ \):
\[
110^\circ + \angle OPS = 180^\circ
\]
Solving for \( \angle OPS \):
\[
\angle OPS = 180^\circ - 110^\circ = 70^\circ
\]

Therefore, the measure of \( \angle OPS \) is \( 70^\circ \).

Would you like further clarification or more details on how this solution was derived?

Here are 5 related questions:
1. What are alternate interior angles and how are they used in geometry?
2. Can we apply the same rule if the lines weren't parallel? Why or why not?
3. What are the properties of transversals intersecting parallel lines?
4. How do you identify corresponding angles in parallel lines?
5. How do linear pairs of angles relate to supplementary angles?

**Tip:** When working with parallel lines and transversals, always look for angle relationships like alternate interior, corresponding, and same-side interior angles.

If OQ and RT are parallel lines and m∠RSP = 110°, what is m∠OPS?

In this geometry problem, learn how to find the measure of angle OPS using the same-side interior angles theorem, given parallel lines OQ and RT, and m∠RSP = 110°. This solution is suitable for students in Grade 8 and covers key concepts in angles formed by transversals intersecting parallel lines.

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