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The image shows a circle with points \(A\), \(B\), \(C\), and \(O\), where \(O\) is the center of the circle. You are given the following information:

- \(\angle AOB = 40^\circ\) (the central angle subtended by the chord \(AB\)),
- The goal is to determine the measure of \(\angle ACB\), which is the inscribed angle subtended by the same chord \(AB\).

### Solution:

Using the theorem that relates central angles to inscribed angles, we know that the inscribed angle is half of the central angle subtended by the same arc. Therefore,

\[
\angle ACB = \frac{1}{2} \times \angle AOB
\]

Substituting the given value:

\[
\angle ACB = \frac{1}{2} \times 40^\circ = 20^\circ
\]

Thus, \(\angle ACB = 20^\circ\).

Would you like further details on this solution, or do you have any additional questions?

Here are 5 related questions to explore further:
1. What is the relationship between central and inscribed angles in a circle?
2. How would you calculate the angle at the center if you are given an inscribed angle?
3. What other properties do inscribed angles in a circle have?
4. How do angles subtended by the same arc relate to each other?
5. How can you use the concept of a cyclic quadrilateral to solve angle problems?

**Tip:** In circle geometry, the angle subtended by a chord at the center is always twice the angle subtended by the same chord at any point on the circle’s circumference.

Apollonius's theorem: Given a circle with points A, B, C, and O (center of the circle), determine the measure of ∠ACB if ∠AOB = 40°.

Learn how to find the measure of ∠ACB in a circle where ∠AOB = 40°. This solution explores key circle geometry concepts, including the Inscribed Angle Theorem, suitable for students in Grades 9-11.

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