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.

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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.

1

020 040 040030 060 0600x 02x 02x

What does the table show about the relationship between and ?

1m mPK

11 2m mPK

Inscribed Angle TheoremThe measure of an angle inscribed

in a circle is equal to ½ its intercepted arc.

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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.

0130

0110

050

070

0650650350350500900120

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

0120 030

0

0

0

90

90

90 0600100 040 050080 050 040

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.

075

083

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