1. Arcs and Chords lesson 10.2 California State Standards
4: Prove theorems involving congruence and similarity
7: Prove/use theorems involving circles.
16: Find probability given graph or table
21: Prove/solve relationships with circles.

2. definitions Central Angle Minor Arc Major Arc Semicircle
An angle whose vertex is the center of a circle.
Minor Arc
A section of the circle cut by a central angle that
measures less than 180o.
Major Arc
A section of the circle that measures more than 180o.
Semicircle
An arc with endpoints coinciding with the endpoints
of a diameter

3. definitions A
minor arc
major arc
central angle C B

4. definitions
semicircle C

5. definitions Measure of a minor arc Measure of a major arc
Equals the measure of its central angle.
Measure of a major arc
Equals the difference between 360o and
the measure of its associated minor arc. A C D B

6. definitions because Congruent Arcs
Arcs with the same measure within the same
circle or congruent circles. E A
because C D B

7. postulate Arc Addition Postulate
The measure of an arc formed by two adjacent
arcs is the sum of the measures of the two arcs. Q P C R

8. example
Find the measure of each arc.
32o E
328o C
180o
148o H I

9. example 142o 218o 118o Find the measure of each arc. S D C T E 60o 82o

10. example
Find the measure of each arc.
60o A B
60o
60o C
yes E D

11. examples No 150o 150o Find the measure of each arc. The circles areB
150o C
210o No D
The circles are
not congruent E

12. examples No 150o 150o Find the measure of each arc. The circles areB
150o C
210o No D
The circles are
not congruent E
homework

13. theorem Congruent Arcs and Chords
Two minor arcs are congruent if and only if
their corresponding chords are congruent. P C B Q A

14. theorem Congruent Arcs and Chords
Two minor arcs are congruent if and only if
their corresponding chords are congruent. P C B Q A

15. theorem Diameter-Chord A diameter that is perpendicular to a chord
bisects the chord and its arc. B D E A C

16. theorem Diameter-Chord Converse
If one chord is the perpendicular bisector of another chord,
then the first chord is a diameter. B D E A C F

17. theorems Congruent Chords Two chords are congruent if and only if
they are equidistant from the center. P R C B Q D A

18. example B
(3x + 11)o
(2x + 48)o C E D

19. examples B C E
(x + 78)o
4xo D

20. example
AB = 12
DE = 12
CE = 7
Find CG 6 B G A D C 7 F E

21. examples Find the center of the circle. Draw two chords
Construct the perpendicular bisector of each chord.

22. example Find the center of the circle. Draw two chords
Construct the perpendicular bisector of each chord.

23. examples Find the center of the circle. Draw two chords
Construct the perpendicular bisector of each chord.
The perpendicular bisectors are diameters.
Diameters intersect at the circle’s center.

24. LOGICAL REASONING What can you conclude about the diagram?
State a postulate or theorem that justifies your answer.
MEASURING ARCS AND CHORDS Find the measure of the red arc or chord in circle A. Explain your reasoning.
MEASURING ARCS AND CHORDS Find the value of x in circle C. Explain your
reasoning.

25. 49. What is the arc measure for each time zone on the wheel?
TIME ZONE WHEEL In Exercises 49–51, use the following information.
The time zone wheel shown at the right consists of two concentric circular pieces of cardboard fastened at the center so the smaller wheel can rotate. To find the time in Tashkent when it is 4 P.M. in San Francisco, you rotate the small wheel until 4 P.M. and San Francisco line up as shown. Then look at Tashkent to see that it is 6 A.M. there. The arcs between cities are congruent.
49. What is the arc measure for each
time zone on the wheel?
50. What is the measure of the minor
arc from the Tokyo zone to the
Anchorage zone?
51. If two cities differ by 180° on the wheel, then it is
3:00 P.M. in one city if and only if it is ____ in the other city.

26. PROVING THEOREM 10.4 In Exercises 56 and 57, you will prove Theorem 10.4 for the case in which the two chords are in the same circle.

27. PROVING THEOREM 10.4 In Exercises 56 and 57, you will prove Theorem 10.4 for the case in which the two chords are in the same circle.

28. PROVING THEOREM 10.7 Write a proof.

29. UNDERSTANDING THE CONCEPT Determine whether the arc is a minor arc, a major arc, or a semicircle of circle R.

30. 35. Name two pairs of congruent arcs in Exercises 32–34.
FINDING ARC MEASURES Find the measure of the red arc.
35. Name two pairs of congruent arcs in Exercises 32–34.
Explain your reasoning.