Geometry: Angles of Circles 1. Compare and contrast central angles and inscribed angles. 2. Explain how the two types of angles (above) are related if they share a common intercepted arc.
A central angle of a circle has its vertex at the center of the circle. The measure of a central angle is defined to be the measure of the intercepted arc.
An inscribed angle has its vertex on the circle (hence in (in or on)-scribed (drawn or written)). The measure of an inscribed angle is 1/2 the measure of the intercepted arc.
If a central angle and an inscribed angle intercept the same arc, the inscribed angle has measure 1/2 the central angle. (The proof requires 3 cases: if the 2 angles have a coincident side (the vertex for the central angle is on a side of the inscribed angle), if the vertex of the central angle is "inside" the inscribed angle, and if the vertex of the central angle is "outside" the inscribed angle.)