Inscribed Angles

Inscribed angles are formed from two chords in a circle, as shown below as angle ABC. The vertex (or where the two chords are) has to touch somewhere on the circumference of the circle.

The measure of the angle is half of the intercepted arc, which is angle AOC in the picture on the right.

Any two inscribed angles with the same intercepted arc are congruent, as shown on the right. This is called the inscribed arc theorem.

Let’s do some practice problems.

We know that the inscribed angle (angle PQR) is half of the angle of the intercepted arc (POR) because of the inscribed angle theorem.

Therefore, angle PQR is 45 degrees, since half of 90 degrees is 45 degrees.

Let’s try a different example, concerning the congruency of the inscribed angles.

The inscribed angles LMN and LPN both have the same intercepted arc LN, so both angles are the same value of 55 degrees.

Great! Now let’s try a quiz to test our knowledge.