Geometry Section 10.3 Inscribed Angles

Recall that a *central angle is an angle What is the relationship between a central angle and the are that it cuts off? whose vertex is at the center of the circle and whose sides are radii. The measure of the central angle equals the measure of its intercepted arc.

An *inscribed angle is an angle whose vertex lies on the circle and whose sides are chords.

By doing the following activity, you will be able to determine the relationship between the measure of an inscribed angle and the measure of its intercepted arc. Given the measure of, complete the table. Remember that the radii of a circle are congruent.

What does the table show about the relationship between and ?

Inscribed Angle Theorem If an angle is an inscribed angle, the measure of the angle is equal to ½ the intercepted arc.

Find the value of x in each figure. Q is the center of each circle.

This work suggests the following theorem.

Theorem 10.11: If a quadrilateral is inscribed in a circle (i.e. its vertices lie on the circle) then, its opposite angles are supplementary.