Suppose you have two points on a circle, C and D. When a radius is drawn from the center to each of these points, an angle could be measured at the center of the circle. This is called the central angle. Now if we pick a third point, and place it somewhere on the circle, we could connect the original points to this new point with chords, and again measure the angle. This is what is known as an inscribed angle.

Study the relationship between the central angle and the inscribed angle when I move the points C and D. What did you notice? You likely saw that the inscribed angle is always half of the central angle, and this will always be true. Even if I drag point E to different positions in the circle, the inscribed angle remains half of the central angle. What does the inscribed angle measure when I make the points D and C form a diameter of the circle, or a 180 degree central angle? The inscribed angle measures 90 degrees. Sometimes, the angle measure for an arc is included along the circumference of a circle. When an arc is measured this way, its angle measurement is always equal to the central angle. Consequently, the Inscribed Angle theorem states that the measure of an inscribed angle must be equal to 1/2 of the measure of its intercepted arc. This intercepted arc is the same as our central angle.

Let's try to solve some sample problems using this theorem. In this first problem, we must find the measure of arc KM knowing that the inscribed angle J is 36 degrees. According to the Inscribed Angle Theorem, the inscribed angle is half of the measure of the intercepted arc. Therefore, 36 is equal to half of x, therefore, x must be 72 degrees.

Another example might look like this. An intercepted arc measures 4y degrees, and the inscribed angle measures 60 degrees. Since the inscribed angle is half the measure of the intercepted arc, then 60 equals one half of 4y… Using algebra to solve for y, we find that y must be equal to thirty.

Finally, we must calculate the measure of angle B and the measure of angle E. First, take note that angle B is the inscribed angle of the intercepted arc ED. The inscribed angle C is also associated with the intercepted arc ED. Therefore angle C is congruent to angle B since they are inscribed angles of the same arc. The measure of angle C must be equal to the measure of angle B which is 28 degrees. Similarly, angle D and angle E are both inscribed angles for the same intercepted arc CB, therefore angle E is congruent to angle D and measures 45 degrees.