The image you provided shows a circle with inscribed angles, labeled using the Inscribed Angle Theorem. The important details from the diagram include:

• $\angle PQS = 60^\circ$
• $\angle SRT = 30^\circ$
• There are two unknown angles labeled $a^\circ$ and $b^\circ$.

Key Points:

1. Inscribed Angle Theorem states that an inscribed angle is half of the measure of the intercepted arc. Thus, we can use this to determine the values of $a$ and $b$.

2. From the diagram:

   • $\angle PQS = 60^\circ$ is an inscribed angle. The intercepted arc of $\angle PQS$ is $\overarc{PS}$, which means the arc $PS$ is $120^\circ$ (since the inscribed angle is half of the arc).

   • Similarly, $\angle SRT = 30^\circ$ implies the intercepted arc $\overarc{QT}$ is $60^\circ$.

3. We can now determine:

   • Angle $a^\circ$: Since the intercepted arc for angle $a$ is $\overarc{PT}$, which is the sum of arcs $PS$ and $QT$, the arc $PT = 120^\circ + 60^\circ = 180^\circ$. Therefore, $\angle PRT = \frac{180^\circ}{2} = 90^\circ$, so $a = 90^\circ$.

   • Angle $b^\circ$: The angle $b$ intercepts the arc $\overarc{PR}$, which is $180^\circ$, so $\angle PRS = 90^\circ$ as well. Thus, $b = 90^\circ$.

Summary:

• $a = 90^\circ$
• $b = 90^\circ$

Would you like further details on how the Inscribed Angle Theorem applies or any other clarification?

Related Questions:

1. How does the Inscribed Angle Theorem relate to the central angle theorem?
2. Can you calculate the arc length if the circle's radius is given?
3. What is the difference between a central angle and an inscribed angle?
4. How would the values of $a$ and $b$ change if the circle was larger but the angles remained the same?
5. How can we use the Inscribed Angle Theorem to solve real-world problems involving circles?

Tip: Remember that the intercepted arc of an inscribed angle is always double the measure of the angle itself!