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
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
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
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
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