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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