How is the central angle of a circle defined?

The central angle of a circle is defined as the angle whose vertex is at the center of the circle and that intercepts an arc on the circle. This means that the two sides of the angle, which are represented by the radii, extend from the center of the circle to points on the circumference, creating an angle that directly correlates to the arc it intercepts.

This definition is significant because it helps in understanding relationships within a circle, such as the fact that the measure of the central angle is equal to the measure of the arc that it intercepts. Recognizing that the vertex must be at the center distinguishes a central angle from other types of angles associated with circles, such as inscribed angles, which have their vertices on the circumference.

The relationship between a central angle and its intercepted arc is a foundational concept in circle geometry, helping students solve various problems involving angles and arcs in circles.