Lesson 9-3 Arcs and Central Angles (page 339) Essential Question How can relationships in a circle allow you to solve problems involving arcs and central angles?

Arcs and Central Angles X B Q Y

CENTRAL ANGLE: an angle with its vertex at the center of a circle. ∠AQX , ∠AQB , ∠AQY , ∠XQB , ∠BQY , ∠XQY examples: A X B Q Y

ARC: an unbroken part of a circle . MINOR ARC examples: , , , , A X B Q Y

ARC: an unbroken part of a circle . MAJOR ARC examples: , , , , Middle letter gives the direction of the arc. A X B Q Y

SEMICIRCLES: if the endpoints of a minor arc are on a diameter . examples: and Middle letter gives the direction of the arc. You must use 3 letters! A Yes, there are two semi-circles! X B Q Y

MEASURE of a MINOR ARC: equals the measure of its central angle MEASURE of a MINOR ARC: equals the measure of its central angle. The measure of any minor arc is less than 180º . 90º A X B Q Y

MEASURE of a MAJOR ARC: equals 360º minus the measure of its minor arc MEASURE of a MAJOR ARC: equals 360º minus the measure of its minor arc. The measure of any major arc is between 180º and 360º . 90º 270º A X B Q Y

MEASURE of a SEMICIRCLE: equals 180º . X B Q Y

ADJACENT ARCS: (of a circle) are arcs with exactly one point in common. example: ________ = __________ Y X Z W

Postulate 16 Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs.

In E, find the measure of the angle or the arc named. Example #1 In E, find the measure of the angle or the arc named. 80º A 70º E 1 B 80º D 80º 80º C

In E, find the measure of the angle or the arc named. Example #2 In E, find the measure of the angle or the arc named. m∠1 = ________ 70º A 70º 70º E 1 B 80º D 80º 80º C

In E, find the measure of the angle or the arc named. Example #3 In E, find the measure of the angle or the arc named. 150º 70º + 80º A 70º 70º E B 80º D 80º 80º C

In E, find the measure of the angle or the arc named. Example #4 In E, find the measure of the angle or the arc named. 290º 360º - 70º A 70º 70º E B 80º D 80º 80º C

CONGRUENT ARCS: arcs in the same circle or in CONGRUENT ARCS: arcs in the same circle or in congruent circles that have equal measures. example: ________ ≅ __________ A 70º 70º E B 80º D 80º 80º C

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. A D 1 2 B C

If ∠1 ≅ ∠2, then ______ ≅ ______ A D 1 2 B C

Written Exercises on pages 341 & 342 GRADED: 1 to 8 all numbers and Assignment Written Exercises on pages 341 & 342 GRADED: 1 to 8 all numbers and 17 to 20 all numbers See the example on page 340 for HELP on #’s17 to 20! How can relationships in a circle allow you to solve problems involving arcs and central angles?

#17 Milwaukee 43ºN 90º rcircle = ? 43ºN 6400km 0º rearth = 6400km

#17 Milwaukee 43ºN 90º rcircle = ? 43º 6400km 90º - 43º = 47º 43º 0º rearth = 6400km