Arcs and Central Angles Section 9-3 Arcs and Central Angles

Central Angle A central angle of a circle has its vertex in the center of the circle . <1 is a central angle in P 1 P

Arcs An arc is an unbroken section of a circle m AB = 88˚ The measure of an arc is equal to the measure of its central angle A B 88˚ m AB = 88˚ A circle has a total of 360˚

Minor and Major Arcs A minor arc is less than 180˚ A major arc is more than 180˚ An arc that is exactly 180˚ is a semicircle D B G E A F H AB FGH CDE C

It’s like… A circle that has been cut in half.

Example 1 C If CA is a diameter of circle O: Name: Two minor arcs Two major arcs Two semicircles An acute central angle Two arcs with the same measure R O A S

Adjacent Arcs Adjacent arcs are arcs that have exactly one endpoint in common and share no other points B C A

Arc Addition Postulate (postulate 16) The measure of the arc formed by two adjacent arcs is the sum of the measures of these arcs. B m AB + m BC = m ABC 64˚ 140˚ Example: C m ABC = 204˚ A

Example 2 Give the measure of each angle or arc if YT is a diameter of circle O WX <WOT XYT Y X 30˚ Z O W 50˚ T

Congruent Arcs Two arcs are congruent if they have the same measure, and if they lie on the same circle or two congruent circles. C 67˚ B O D 67˚ A

Congruent Arcs Two arcs are congruent if they have the same measure, and if they lie on the same circle or two congruent circles. P B 75˚ O 8 cm F 8 cm D 75˚ I

Theorem 9-3 In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent. Conclusion: m<1 = m<2 P B 2 75˚ 1 T F 8 cm 8 cm A 75˚ I

Example 3 Find the measure of central <1 72˚ 95˚ 150˚ 50˚ 1.) 2.) 40˚ 1 1 72˚ 225˚ 3.) 150˚ 4.) 50˚ 1 1 130˚ 30˚

Example 4 D Find the measure of each arc. 1.) AB 2.) BC 3.) CD 4.) DE

Or is it? The End