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To solve the problem, we need to find the measure of the angle \( \angle ACB \) in the circle where the arc \( \overset{\frown}{AB} \) measures \( 260^\circ \).

In a circle, the angle formed by two points on the circle, measured at any point other than the center, is half the measure of the arc between those two points. This rule applies to the inscribed angle theorem.

Given that the arc \( \overset{\frown}{AB} \) measures \( 260^\circ \):
- \( \angle ACB \) is an inscribed angle that subtends arc \( \overset{\frown}{AB} \).

Therefore, the measure of \( \angle ACB \) can be calculated using the inscribed angle theorem:
\[
\angle ACB = \frac{1}{2} \times \overset{\frown}{AB}
\]
\[
\angle ACB = \frac{1}{2} \times 260^\circ = 130^\circ
\]

Thus, \( \angle ACB \) measures \( 130^\circ \).

Would you like any further details or clarification on how this was calculated?

**Further Explorations:**
1. How does the center angle \( \angle AOB \) compare to the inscribed angle \( \angle ACB \)?
2. What would the measurement of \( \angle ACB \) be if arc \( \overset{\frown}{AB} \) were less than \( 180^\circ \)?
3. How can you determine the length of arc \( \overset{\frown}{AB} \) if you know the radius of the circle?
4. What is the effect of the location of point C on the measurement of \( \angle ACB \)?
5. How do properties of a circle change if it is not a circle but an ellipse?

**Tip:** Always remember that the inscribed angle theorem is a powerful tool in solving many circle-related problems, as it relates the angle inscribed in the circle to the arc it subtends.

Learn how to calculate the inscribed angle ACB in a circle where the arc AB measures 260°. This problem demonstrates the use of the inscribed angle theorem and its application to circle geometry. Suitable for students in Grades 9-12.

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