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

2. Arcs and Central AnglesX B Q Y

3. 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

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

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

6. 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

7. 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

8. 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

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

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

11. 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.

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. If ∠1 ≅ ∠2, then ______ ≅ ______A D 1 2 B C

19. Written Exercises on pages 341 & 342
Assignment
Written Exercises on pages 341 & 342
1 to 8 and 16 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 segments, angles, and arcs?

20. #17
Milwaukee
43ºN
90º
rcircle = ?
43ºN
6400km 0º
rearth = 6400km

21. #17 Milwaukee 43ºN 90º rcircle = ? 43º 6400km 90º - 43º = 47º 43º 0º
rearth = 6400km

22. Written Exercises on pages 341 & 342
Assignment
Written Exercises on pages 341 & 342
RECOMMENDED: 1, 2, 3, 4, 5, 6, 8
REQUIRED: 7, 16, 17, 18, 19, 20
~ BONUS: #23 ~
How can relationships in a circle allow you to solve problems involving segments, angles, and arcs?

23. WORK leading to the final answer.
~ Bonus Assignment ~
#23 from page 343.
Include diagram and
WORK leading to the final answer.
Worth 10 points.
How can relationships in a circle allow you to solve problems involving segments, angles, and arcs?