In the image provided, you have a problem that involves using the Inscribed Angle Theorem to find the values of $a^\circ$ and $b^\circ$ in a circle diagram. Here’s how we approach it:

Inscribed Angle Theorem:

This theorem states that the measure of an inscribed angle is half the measure of the intercepted arc.

Step-by-Step Solution:

1. For $a^\circ$ (angle $\angle PQT$):

   • Angle $\angle PQT$ subtends the arc $PR$.
   • Based on the inscribed angle theorem, $\angle PQT = \frac{1}{2} \times \text{arc } PR$.
   • The arc $PR$ subtends an angle at the center $O$, which is $\angle POR = 60^\circ$ (since the angle at $P$ is given as $60^\circ$).

   Therefore, $a = \frac{1}{2} \times 60^\circ = 30^\circ.$

2. For $b^\circ$ (angle $\angle QRS$):

   • Angle $\angle QRS$ subtends the arc $PS$, which is the entire arc from $P$ to $S$.
   • We know that the angle $\angle PTS$ is given as $30^\circ$, which means this inscribed angle subtends the same arc $PS$.
   • Therefore, by the Inscribed Angle Theorem:

   $b = 30^\circ.$

Summary of the Results:

• $a^\circ = 30^\circ$
• $b^\circ = 30^\circ$

Would you like further clarification on this, or do you have any questions?

Here are some additional related questions:

1. How does the Inscribed Angle Theorem apply to other types of circle segments?
2. What would change if the central angle was given instead of the inscribed angle?
3. How can we calculate arc lengths using the inscribed angles?
4. What happens if two inscribed angles subtend the same arc?
5. How do the properties of inscribed angles change if the circle's radius changes?

Tip: Always check whether an angle subtends a major or minor arc before applying the inscribed angle theorem!