Post on 30-Dec-2015
description
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Lesson 9-3
Arcs and Central Angles
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Central Angle
(of a circle)
Central Angle
(of a circle)
NOT A Central Angle
(of a circle)
Central AngleAn angle whose vertex lies on the center of the circle.Definition:
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Central Angle TheoremThe measure of a center angle is equal to the measure of the intercepted arc.
AD
Y
Z
O 110
110
Intercepted Arc Center Angle
Example: is the diameter, find the value of x and y and z in the figure.
z°
25°
55°y°
x°
O
B
D
AC
25
180 (25 55 ) 180 80 100
55
x
y
z
=
= − + = − =
=
o
o o o o
o
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Example: Find the measure of each arc.
3x+10
2x-14
2x4x
3x
B
D
C
E
A
4x + 3x + (3x +10) + 2x + (2x-14) = 360°
14x – 4 = 360°14x = 364°
x = 26°
4x = 4(26) = 104°
3x = 3(26) = 78°
3x +10 = 3(26) +10= 88°
2x = 2(26) = 52°
2x – 14 = 2(26) – 14 = 38°
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Measures of an ArcMeasures of an Arc:
1. The measure of a minor arc is the measure of its central angle. 2. The measure of a major arc is 360 - (measure of its minor arc). 3. The measure of any semicircle is 180.
Adjacent Arcs:
Arcs in a circle with exactly one point in common.
A
B
C
T
P
List: Major Arcs Minor Arcs Semicircles Adjacent Arcs
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Theorem• In the same circle or in congruent circles, two
minor arcs are congruent if and only if their central angles are congruent.
M
N
R
S
O
If ∠RNO =∠SNO, then ROª =OSª
If ROª =OSª , then∠RNO =∠SNO
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Arc Addition Postulate• The measure of the arc formed by two
adjacent arcs is the sum of the measures of these two arcs.
H
J G
K
F
HGº +GFª =HFª
soif HGº =50oandGFª =55o,
then HFª =105o
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Example• In circle J, find the measures
of the angle or arc named with the given information:
• Find:
H
J G
K
F
HGº =70
KFª =80∠GJF =80
GFª
HFª
∠HJG
HKGº
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In Circle C, find the measure of each arc or angle named.
• Given: SP is a diameter of the circle. Arc ST = 80 and Arc QP=60.
• Find:
T
C
P
S
Q
SQª
SPQº
SPT
SPª
∠SCQ
º