Section 10-4, 10-6 spi.3.3.A, spi.3.3.B Section 10-4, 10-6 spi.3.3.A, spi.3.3.B

Jim Smith JCHSJim Smith JCHS

Central Angles Are Equal To The Central Angles Are Equal To The Measure Of The Intercepted ArcMeasure Of The Intercepted Arc

Central AnglesCentral Angles

1163°63°

1 = 63°1 = 63°

Inscribed AnglesInscribed Angles

The Vertex Of Inscribed Angles AreThe Vertex Of Inscribed Angles AreOn The Circle And The Sides AreOn The Circle And The Sides Are

Contained In Chords. Contained In Chords.

11 224433

The Measure Of An Inscribed AngleThe Measure Of An Inscribed Angle

Is Equal To Is Equal To ½½ The Measure Of The The Measure Of TheIntercepted ArcIntercepted Arc

11

100°100°

1 = ½ ( 100 )1 = ½ ( 100 )1 = 50°1 = 50°

224433

2 = ½ (180 )2 = ½ (180 )2 = 90°2 = 90°

Any Angle With It’s VertexAny Angle With It’s Vertex On The Circle Is Equal On The Circle Is Equal

To ½ Of The Arc.To ½ Of The Arc.

2 secants2 secantsA secant andA secant and a tangenta tangent

If The Vertex Is If The Vertex Is SomewhereSomewhereInside The Circle But Not At TheInside The Circle But Not At TheCenter, The Angle Is Equal ToCenter, The Angle Is Equal To

½½ The The Sum Sum Of The ArcsOf The Arcs

88

150°150°

80°80°

8 = ½ (150 + 8 = ½ (150 + 80)80)

8 = ½ ( 230 )8 = ½ ( 230 )

8 = 115°8 = 115°

If The Vertex Is OutsideIf The Vertex Is Outside(A (A DistanceDistance Away From) The Away From) The

Circle, The Arc Is Equal ToCircle, The Arc Is Equal To

½½ The The DifferenceDifference Of The Arcs. Of The Arcs.

120°120°50°50°

55

5 = ½ (120 - 50)5 = ½ (120 - 50)5 = ½ ( 70 )5 = ½ ( 70 )5 = 35°5 = 35°

2211

3344

= arc= arc =1/2 arc =1/2 arc

=1/2 (sum of =1/2 (sum of 2 arcs)2 arcs)

=1/2 (difference=1/2 (difference of 2 arcs)of 2 arcs)

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AC is a diameterAC is a diameter

CD = 100°CD = 100°

CB = AECB = AE

BE = 50°BE = 50° 1122

33

44

55Find theFind theMeasures ofMeasures ofthe anglesthe angles

AF is a tangentAF is a tangent

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AA

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AC is a diameterAC is a diameter

CD = 100°CD = 100°

CB = AECB = AE

BE = 50°BE = 50° 1122

33

44

55

AF is a tangentAF is a tangent

100°100° 80°80°

50°50°

130/2 =65130/2 =65

65°65°

65°65°

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1122

33

44

55

100°100° 80°80°

50°50°65°65°

65°65°

1 = 80°1 = 80°

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FFEE

1122

33

44

55

100°100° 80°80°

50°50°65°65°

65°65°

2 =1/2 (65)2 =1/2 (65)

2 = 32 ½ °2 = 32 ½ °

3 =1/2 (180+50)3 =1/2 (180+50) 3 =1/2 (230)3 =1/2 (230) 3 = 115°3 = 115°

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FFEE

1122

33

44

55

100°100° 80°80°

50°50°65°65°

65°65°

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1122

33

44

55

100°100° 80°80°

50°50°65°65°

65°65°

4 =1/2 (65+50)4 =1/2 (65+50)

4 =1/2 (115)4 =1/2 (115)

4 = 57 ½ °4 = 57 ½ °AA

5 =1/2 (180-65)5 =1/2 (180-65)

5 =1/2 (115)5 =1/2 (115)

5 =57 1/2°5 =57 1/2°

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1122

33

44

55

100°100° 80°80°

50°50°65°65°

65°65°