Properties of Circles

1. Section 5-8
Properties of Circles

2. Essential Questions
• What are the relationships among parts of
a circle?
• What are the properties of circles and how
do you apply them?
• Where you’ll see this:
• Market research, food service, art,
recreation, navigation

3. Vocabulary
1. Circle:
2. Radius:
3. Chord:
4. Diameter:
5. Central Angle:

4. Vocabulary
1. Circle: All points that are the same distance from a
fixed center point; 360° total
2. Radius:
3. Chord:
4. Diameter:
5. Central Angle:

5. Vocabulary
1. Circle: All points that are the same distance from a
fixed center point; 360° total
2. Radius: A segment whose endpoints are the center
of a circle and on the circle
3. Chord:
4. Diameter:
5. Central Angle:

6. Vocabulary
1. Circle: All points that are the same distance from a
fixed center point; 360° total
2. Radius: A segment whose endpoints are the center
of a circle and on the circle
3. Chord: A segment where both endpoints are on the
circle
4. Diameter:
5. Central Angle:

7. Vocabulary
1. Circle: All points that are the same distance from a
fixed center point; 360° total
2. Radius: A segment whose endpoints are the center
of a circle and on the circle
3. Chord: A segment where both endpoints are on the
circle
4. Diameter: A chord that goes through the center of a
circle
5. Central Angle:

8. Vocabulary
1. Circle: All points that are the same distance from a
fixed center point; 360° total
2. Radius: A segment whose endpoints are the center
of a circle and on the circle
3. Chord: A segment where both endpoints are on the
circle
4. Diameter: A chord that goes through the center of a
circle
5. Central Angle: An angle where the vertex is the
center of the circle

9. Vocabulary
6. Arc:
7. Semicircle:
8. Minor Arc:
9. Major Arc:
10. Inscribed Angle:

10. Vocabulary
6. Arc: A section of the circumference of a circle
7. Semicircle:
8. Minor Arc:
9. Major Arc:
10. Inscribed Angle:

11. Vocabulary
6. Arc: A section of the circumference of a circle
7. Semicircle: An arc that is half of the circumference;
half a circle
8. Minor Arc:
9. Major Arc:
10. Inscribed Angle:

12. Vocabulary
6. Arc: A section of the circumference of a circle
7. Semicircle: An arc that is half of the circumference;
half a circle
8. Minor Arc: An arc that is less than half the
circumference; same measure as the central angle
9. Major Arc:
10. Inscribed Angle:

13. Vocabulary
6. Arc: A section of the circumference of a circle
7. Semicircle: An arc that is half of the circumference;
half a circle
8. Minor Arc: An arc that is less than half the
circumference; same measure as the central angle
9. Major Arc: An arc that is more than half the
circumference
10. Inscribed Angle:

14. Vocabulary
6. Arc: A section of the circumference of a circle
7. Semicircle: An arc that is half of the circumference;
half a circle
8. Minor Arc: An arc that is less than half the
circumference; same measure as the central angle
9. Major Arc: An arc that is more than half the
circumference
10. Inscribed Angle: An angle whose vertex is on the
circle and whose sides are chords of the circle; half
the measure of the arc it contains

15. Circle

16. Radius

17. Chord

18. Diameter

19. Central Angle

20. Arc

21. Semicircle

22. Minor Arc

23. Major Arc

24. Inscribed Angle

25. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫

26. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
132°

27. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
132° 82°

28. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
132° 82°
x°

29. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
132° 82°
x° x°

30. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
x + x +132 + 82 = 360
132° 82°
x° x°

31. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
x + x +132 + 82 = 360
132° 82°
2x + 214 = 360
x° x°

32. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
x + x +132 + 82 = 360
132° 82°
2x + 214 = 360
−214 −214
x° x°

33. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
x + x +132 + 82 = 360
132° 82°
2x + 214 = 360
−214 −214
2x =146
x° x°

34. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
x + x +132 + 82 = 360
132° 82°
2x + 214 = 360
−214 −214
2x =146
2 2
x° x°

35. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
x + x +132 + 82 = 360
132° 82°
2x + 214 = 360
−214 −214
2x =146
2 2
x° x° x = 73

36. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
x + x +132 + 82 = 360
132° 82°
2x + 214 = 360
−214 −214
2x =146
2 2
73° 73° x = 73

37. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
132° 82°
73° 73°

38. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ª ª )
132° 82° m∠ABC = (mAD + mCD
2
73° 73°

39. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ª ª )
132° 82° m∠ABC = (mAD + mCD
2
1
= (73 + 73)
2
73° 73°

40. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ª ª )
132° 82° m∠ABC = (mAD + mCD
2
1 1
= (73 + 73) = (146)
2 2
73° 73°

41. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ª ª )
132° 82° m∠ABC = (mAD + mCD
2
1 1
= (73 + 73) = (146) = 73°
2 2
73° 73°

42. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
132° 82°
73° 73°

43. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ª ª )
132° 82° m∠BCD = (mAD + mAB
2
73° 73°

44. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ª ª )
132° 82° m∠BCD = (mAD + mAB
2
1
= (73 +132)
2
73° 73°

45. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ª ª )
132° 82° m∠BCD = (mAD + mAB
2
1 1
= (73 +132) = (205)
2 2
73° 73°

46. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ª ª )
132° 82° m∠BCD = (mAD + mAB
2
1 1
= (73 +132) = (205) =102.5°
2 2
73° 73°

47. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
132° 82°
73° 73°

48. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ∫ ª )
132° 82° m∠CDA = (mBC + mAB
2
73° 73°

49. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ∫ ª )
132° 82° m∠CDA = (mBC + mAB
2
1
= (82 +132)
2
73° 73°

50. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ∫ ª )
132° 82° m∠CDA = (mBC + mAB
2
1 1
= (82 +132) = (214)
2 2
73° 73°

51. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ∫ ª )
132° 82° m∠CDA = (mBC + mAB
2
1 1
= (82 +132) = (214) =107°
2 2
73° 73°

52. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
132° 82°
73° 73°

53. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ∫ ª )
132° 82° m∠DAB = (mBC + mCD
2
73° 73°

54. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ∫ ª )
132° 82° m∠DAB = (mBC + mCD
2
1
= (82 + 73)
2
73° 73°

55. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ∫ ª )
132° 82° m∠DAB = (mBC + mCD
2
1 1
= (82 + 73) = (155)
2 2
73° 73°

56. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
1 ∫ ª )
132° 82° m∠DAB = (mBC + mCD
2
1 1
= (82 + 73) = (155) = 77.5°
2 2
73° 73°

57. Example 1
ª ≅ CD . Find the measures of the
ª
In circle O, AD
angles of quadrilateral ABCD, when
ª =132° and mBC = 82°.
mAB ∫
132° 82° m∠ABC = 73°
m∠BCD =102.5°
m∠CDA =107°
73° 73°
m∠DAB = 77.5°

58. Example 2
Identify the following for circle P.
a. Diameter b. Radius
c. Chord ª
d. mLM
)
º
e. mLMK f. mLJ
g. m∠LKJ h. Central Angle

59. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK
c. Chord ª
d. mLM
)
º
e. mLMK f. mLJ
g. m∠LKJ h. Central Angle

60. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK KP
c. Chord ª
d. mLM
)
º
e. mLMK f. mLJ
g. m∠LKJ h. Central Angle

61. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK KP
c. Chord ª
d. mLM
KL
)
º
e. mLMK f. mLJ
g. m∠LKJ h. Central Angle

62. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK KP
c. Chord ª
d. mLM
KL = 62° + 47°
)
º
e. mLMK f. mLJ
g. m∠LKJ h. Central Angle

63. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK KP
c. Chord ª
d. mLM
KL = 62° + 47° =109°
)
º
e. mLMK f. mLJ
g. m∠LKJ h. Central Angle

64. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK KP
c. Chord ª
d. mLM
KL = 62° + 47° =109°
)
º
e. mLMK f. mLJ
= 62° +180°
g. m∠LKJ h. Central Angle

65. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK KP
c. Chord ª
d. mLM
KL = 62° + 47° =109°
)
º
e. mLMK f. mLJ
= 62° +180° = 242°
g. m∠LKJ h. Central Angle

66. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK KP
c. Chord ª
d. mLM
KL = 62° + 47° =109°
)
º
e. mLMK f. mLJ
= 62° +180° = 242° = 62°
g. m∠LKJ h. Central Angle

67. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK KP
c. Chord ª
d. mLM
KL = 62° + 47° =109°
)
º
e. mLMK f. mLJ
= 62° +180° = 242° = 62°
g. m∠LKJ h. Central Angle
1
= 2 (62°)

68. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK KP
c. Chord ª
d. mLM
KL = 62° + 47° =109°
)
º
e. mLMK f. mLJ
= 62° +180° = 242° = 62°
g. m∠LKJ h. Central Angle
= 2 (62°) = 31°
1

69. Example 2
Identify the following for circle P.
a. Diameter b. Radius
JK KP
c. Chord ª
d. mLM
KL = 62° + 47° =109°
)
º
e. mLMK f. mLJ
= 62° +180° = 242° = 62°
g. m∠LKJ h. Central Angle
= 2 (62°) = 31°
1
∠JPM

70. Homework

71. Homework
p. 228 #1-25 odd
“We are so accustomed to disguise ourselves to others
that in the end we become disguised to ourselves.”
- Francois de La Rochefoucauld