Arcs and Chords lesson 10.2

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

4. semicircle definitions C

5. definitions • Measure of a minor 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 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 148o 180o H I

9. example Find the measure of each arc. 142o S D 218o 60o 82o C 118o 100o T E

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

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

12. examples Find the measure of each arc. A 150o B 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 (2x + 48)o (3x + 11)o C D E

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. 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.

22. 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.

23. 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.

24. 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.

25. PROVING THEOREM 10.7 Write a proof.