Let's solve the problem by analyzing each angle or arc measure requested:

Given Information

• $m\angle P = 57^\circ$
• Arc $PS$ is $137^\circ$
• Arc $RQ$ is $41^\circ$

Solution Steps

1. Finding $m\angle Q$:

   • Angle $Q$ is an inscribed angle subtended by arc $PS$.
   • The measure of an inscribed angle is half the measure of the arc it intercepts.
   • Therefore, $m\angle Q = \frac{137}{2} = 68.5^\circ$.

2. Given $m\angle R = 123^\circ$:

   • This angle measure is provided, so we do not need to calculate it.

3. Finding $m\angle S$:

   • Angle $S$ is an inscribed angle subtended by arc $RQ$.
   • Similarly, $m\angle S = \frac{41}{2} = 20.5^\circ$.

Final Answers

• $m\angle Q = 68.5^\circ$
• $m\angle R = 123^\circ$
• $m\angle S = 20.5^\circ$

Would you like more detailed explanations, or do you have any specific questions?

Here are some related questions for further exploration:

1. How do you calculate the measure of an inscribed angle in terms of the intercepted arc?
2. What is the relationship between central angles and inscribed angles in a circle?
3. How can we determine the arc length if the radius of the circle is given?
4. What are the properties of opposite angles in a cyclic quadrilateral?
5. How does an inscribed angle change when the intercepted arc changes?

Tip: Always remember that an inscribed angle in a circle is half the measure of the intercepted arc.