The diagram shows two intersecting triangles: $\triangle POQ$ and $\triangle ROS$. You are given three angle measures:

$m\angle P = 65^\circ, \quad m\angle Q = 42^\circ, \quad m\angle R = 50^\circ.$

It seems you are being asked to determine certain angles related to these triangles and possibly the angles formed at the intersection point $O$.

1. Angle $m\angle POQ$:

This angle is part of the triangle $\triangle POQ$. Using the triangle angle sum property:

$m\angle P + m\angle Q + m\angle POQ = 180^\circ.$

Plugging in the given values:

$65^\circ + 42^\circ + m\angle POQ = 180^\circ,$

$m\angle POQ = 180^\circ - 107^\circ = 73^\circ.$

So, the measure of $\angle POQ$ is:

$m\angle POQ = 73^\circ.$

2. Determining $m\angle ROS$:

There is no direct information for this triangle, but if it is a vertically opposite angle to $\angle POQ$, they will be equal. Since vertically opposite angles are congruent, we can conclude:

$m\angle ROS = 73^\circ.$

Final Answer:

$m\angle POQ = 73^\circ, \quad m\angle ROS = 73^\circ.$

Would you like further clarification or help with related questions?

5 Follow-up Questions:

1. What is the sum of all interior angles in any triangle?
2. How do you determine vertically opposite angles?
3. Can two different triangles share identical angle measures?
4. How do adjacent angles relate to each other at a common intersection point?
5. What happens when two angles are complementary or supplementary?

Tip: Always verify vertical and adjacent angle properties carefully when solving geometry problems with intersecting lines or triangles.