Answer:

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 ∠ABC. The other end points than the vertex, A and C define the intercepted arc AC of the circle. The measure of AC is the measure of its central angle. That is, the measure of ∠AOC.

Inscribed Angle Theorem:

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

That is, m∠ABC=12m∠AOC.

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

Here, ∠ADC≅∠ABC≅∠AFC.

Example 1:

Find the measure of the inscribed angle ∠PQR.

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 ∠POR of the intercepted arc PR is 90°.

Therefore,

m∠PQR=12m∠POR                   =12(90°)                    =45°. #ihopeitshelps