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.