https://file.aiquickdraw.com/mathbot/file/1729329657102udvnwvaq.jpg

The problem in the image asks for the measure of angle \( \angle PBC \) in the given circle. The central angle \( \angle AOC \) is given as \( 120^\circ \), and point \( B \) lies on the circle's circumference.

Since \( \angle AOC \) is a central angle, the arc subtended by this angle is also subtended by the inscribed angle \( \angle PBC \) at point \( B \). The property of a circle tells us that the inscribed angle is half the central angle subtended by the same arc.

Therefore, 
\[
\angle PBC = \frac{1}{2} \times \angle AOC = \frac{1}{2} \times 120^\circ = 60^\circ.
\]

So, the measure of \( \angle PBC \) is \( 60^\circ \).

Would you like a detailed explanation of this concept, or do you have any further questions?

Here are 5 related questions:
1. What is the relationship between a central angle and the arc it subtends?
2. How do you find the measure of an inscribed angle in a circle?
3. Can an inscribed angle ever be equal to the central angle subtending the same arc?
4. What is the sum of the angles in a triangle inscribed in a circle?
5. What is the measure of an angle subtended by a semicircle?

**Tip:** In a circle, the inscribed angle is always half of the central angle subtending the same arc.

What is the measure of angle PBC in the following figure?

Learn how to find the measure of angle PBC in a circle using the inscribed angle theorem. Given that the central angle AOC is 120°, this problem shows how to apply the formula to find the inscribed angle. Suitable for Grades 9-10 students studying circle geometry.

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