The Inscribed Angle Theorem and Its Applications

An inscribed angle is an angle formed by two chords of a circle with the vertex on its circumference.  In the first circle in Figure 1, segments AB and AC are chords of a circle and the vertex A is on its circumference. Hence, angle A is an inscribed angle.  In the second circle in Figure 1, angle Q is also an inscribed angle.

Angles formed by the two radii of a circle are called central angles. The vertex of a central angle is on the center of the circle. In the Figure 1 below, angles BOC and SOT are central angles.

The arc determined by the endpoints of two chords or two radii, and is opposite of the angle, is called the intercepted arc. In the first circle in Figure 1, arc BC, denoted by the red curve, is the intercepted arc of angle BAC (or angle BOC). Similarly, in the second circle, arc ST denoted by the green curve, is the intercepted arc of the central angle SOT.  Saying the other way around,  angle BAC is the subtended angle of arc BC and and angle SOT is the subtended angle of arc ST.

The Inscribed Angle Theorem

In this article, we are going to discuss the relationship between an inscribed angle and a central angle (I have created a GeoGebra applet about it) having the same intercepted arc.  This is shown in the first circle in Figure 1. Angle BAC and angle BOC have the same intercepted arc BC. Therefore, if a central and an inscribed angle have the same intercepted arc, the measure of that central angle is twice that of the measure of the inscribed angle.


Figure 1 – Inscribed angles, central angles and their intercepted arcs.


One strategy to solve a problem in geometry is to sometimes draw lines from one point to another. These lines drawn are called auxiliary lines. » Read more