To solve the problem, let's use the Inscribed Angle Theorem. This theorem states that an inscribed angle is half the measure of the intercepted arc. Here's the breakdown:

Step-by-Step Solution

1. Given Information:

   • $\angle QPR = a^\circ$
   • $\angle QRS = 60^\circ$
   • $\angle RTS = 30^\circ$
   • $\angle RQS = b^\circ$

2. Determine $a$ using the Inscribed Angle Theorem:

   • The angle $\angle QPR = a^\circ$ intercepts arc $QR$.
   • $\angle QRS$ intercepts the same arc, and it is $60^\circ$.
   • Since $\angle QPR = \frac{1}{2} \times \text{arc} \, QR$, and $\angle QRS$ also intercepts arc $QR$, $a = 60^\circ.$

3. Determine $b$ using the Inscribed Angle Theorem:

   • The angle $\angle QRS = b^\circ$ intercepts arc $PR$.
   • $\angle RTS = 30^\circ$ intercepts the same arc.
   • Since $\angle QRS = \frac{1}{2} \times \text{arc} \, PR$, and $\angle RTS$ intercepts arc $PR$, $b = 30^\circ.$

Solution:

• $a = 60^\circ$
• $b = 30^\circ$

Would you like a detailed explanation or have any questions? Here are five related questions to explore further:

1. How does the Inscribed Angle Theorem apply to different types of circles?
2. What are central angles, and how do they relate to inscribed angles?
3. Can the Inscribed Angle Theorem be used to determine angles outside of a circle?
4. What is the relationship between opposite angles in a cyclic quadrilateral?
5. How would the problem change if the angles were not given, but only arcs?

Tip: Remember, the Inscribed Angle Theorem is crucial for quickly determining relationships between angles and arcs in circles!