Inscribed Angles

An inscribed angle in a circle is formed by two chords that have a common end point on the circle. This common end point is the vertex of the angle.

Here, the circle with center $O$ has the inscribed angle $\angle ABC$. The other end points than the vertex, $A$ and $C$ define the intercepted arc $\stackrel{⌢}{AC}$ of the circle. The measure of $\stackrel{⌢}{AC}$  is the measure of its central angle. That is, the measure of $\angle AOC$.

Inscribed Angle Theorem:

The measure of an inscribed angle is half the measure of the intercepted arc.

That is, $m\angle ABC=\frac{1}{2}m\angle AOC$.

This leads to the corollary that in a circle any two inscribed angles with the same intercepted arcs are congruent.

Here, $\angle ADC\cong \angle ABC\cong \angle AFC$.

An especially interesting result of the Inscribed Angle Theorem is that an angle inscribed in a semi-circle is a right angle.

In a semi-circle, the intercepted arc measures $180°$ and therefore any corresponding inscribed angle would measure half of it.