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 has the inscribed angle . The other end points than the vertex, and define the intercepted arc of the circle. The measure of is the measure of its central angle. That is, the measure of .
Inscribed Angle Theorem:
The measure of an inscribed angle is half the measure of the intercepted arc.
That is, .
This leads to the corollary that in a circle any two inscribed angles with the same intercepted arcs are congruent.
Here, .
Example 1:
Find the measure of the inscribed angle .
By the inscribed angle theorem, the measure of an inscribed angle is half the measure of the intercepted arc.
The measure of the central angle of the intercepted arc is .
Therefore,
.
Example 2:
Find .
In a circle, any two inscribed angles with the same intercepted arcs are congruent.
Here, the inscribed angles and have the same intercepted arc .
So, .
Therefore, .
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 and therefore any corresponding inscribed angle would measure half of it.
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