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9 – 3 Arcs and Central Angles

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Arcs and Central Angles

A central angle of a circle is an angle with its vertex at the center of the circle.

O

Y

Z

< YOZ is a central angle

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Arcs and Central Angles

An arc is an unbroken part of a circle. Two points y and z on a circle O are always the endpoints of two arcs. Y and Z and the points of circle O in the interior Of <YOZ form a minor arc.

O

Y

Z

Minor ArcYZ

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Arcs and Central Angles

Y and Z and the remaining points of circle

O form a major arc.

O

Y

Z

W

Major Arc

YW Z

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Arcs and Central Angles

If Y and Z are the endpoints of a diameter, then the two arcs are called semicircles. You use 3 points to specify a semicircle or major arc.

Semicircles

OY Z

W

YWZ

OY Z

X

YXZ

Diameter Diameter

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Arcs and Central Angles

The measure of a minor arc is defined to be the measure of its central angle. In the diagram mYZ represents the measure of minor arc YZ.

O

Y

Z

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Arcs and Central Angles

The measure of a major arc is 360 minus the measure of its minor arc. The mYWZ

is 360 - 50 = 310

O

Y

Z

W

Major Arc

YW Z

50

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Arcs and Central Angles

The measure of a semicircle is 180

OY Z

X

YXZ

Diameter

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Arcs and Central Angles

Adjacent arcs of a circle are arcs that that have exactly one point in common. The following postulate can be used to find the measure of an arc formed by two adjacent arcs.

Postulate 16The measure of the arc formed by two

adjacent arcs is the sum of the measure of its arcs.

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Arcs and Central AnglesExample Postulate 16

Applying the ARC Addition Postulate 60 + 50 = 110

60

50

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Arcs and Central Angles

Congruent arcs are arcs, in the same circle or congruent circles that have equal measures.

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Arcs and Central AnglesTheorem 9-3

In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent

P

60

Q

60

N

60

P = Q, therefore 2 minor arcs are congruent

P = Q,Therefore 2 minor arcs are not congruent