How is the central angle defined in relation to the intercepted arc?

The central angle is defined as the angle whose vertex is at the center of a circle and whose sides (or rays) extend to the endpoints of an arc on the circle. In relation to the intercepted arc, the measure of the central angle is directly equal to the measure of the intercepted arc, which is the arc that lies in the interior of the angle.

When measuring angles in circles, the central angle and the intercepted arc both are expressed in degrees. For example, if a central angle measures 60 degrees, the intercepted arc will also measure 60 degrees. This fundamental property is essential in understanding various relationships and theorems related to circles in geometry, including those involving inscribed angles and arc lengths.

The other choices do not accurately represent the relationship between the central angle and the intercepted arc. For instance, while the length of an arc pertains to physical distance along the arc, it does not indicate the measure of the angle in degrees. Similarly, stating the central angle is half the intercepted arc applies to inscribed angles, not central angles. The entire circumference of the circle pertains to the total distance around the circle and is not applicable when defining the relationship of one specific angle to one specific arc.