Geometry Honors Section 9.3 Arcs and Inscribed Angles

Recall that a *central angle is an angle What is the relationship between a central angle and the arc 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 The measure of an angle inscribed in a circle is equal to ½ its intercepted arc.

the angles are congruent. Corollaries of the Inscribed Angle Theorem: If two inscribed angles intercept the same arc, then If an inscribed angle intercepts a semicircle, then the angles are congruent. the angle is a right angle.

A second type of angle that has its vertex on the circle is an angle formed by a tangent and a chord intersecting at the point of tangency.

Theorem: If a tangent and a chord intersect on a circle at the point of tangency, then the measure of the angle formed is equal to ½ the measure of the intercepted arc.