Download 10.2 Warmup x graph. Pay close attention to the units. April 14, 2016

10.2 Warmup
Determine the value of x for the circle
graph. Pay close attention to the units.
April 14, 2016
Geometry 10.2 Finding Arc Measures
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Geometry
10.2 Finding Arc Measures
10.2 Essential Question
How are circular arcs measured?
April 14, 2016
Geometry 10.2 Finding Arc Measures
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Goals
○ Identify arcs & chords in circles
○ Compute arc measures and angle
measures
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Geometry 10.2 Finding Arc Measures
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Central Angle
A
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An angle whose
vertex is the
center of a
circle.
Geometry 10.2 Finding Arc Measures
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Minor Arc
C
A
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Part of a circle.
The measure of
the central
T angle is less
than 180.
CT
Geometry 10.2 Finding Arc Measures
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Semicircle
C
A
D
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Half of a circle. The
endpoints of the arc
are the endpoints of
a diameter. The
central angle
T
measures 180.
CTD
Geometry 10.2 Finding Arc Measures
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Major Arc
C
A
D
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T
Part of a circle.
The measure of
the central
angle is greater
than 180.
CTD
Geometry 10.2 Finding Arc Measures
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Major Arc
CTD
C
BUT NOT
A
D
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T
CDT
Geometry 10.2 Finding Arc Measures
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Measuring Arcs
○ An arc has the same measure as the
central angle.
○ We say, “a central angle subtends an arc
of equal measure”.
mACB  42
A
42
42
C
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B
mAB  42
Geometry 10.2 Finding Arc Measures
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Measuring Major Arcs
○ The measure of a major arc is given by
360  measure of minor arc.
mACB  42
A
42
42
D
C
mAB  42
B
mADB  360  42  318
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Geometry 10.2 Finding Arc Measures
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Arc Addition Postulate
○ The sum of the parts equals the whole.
R
T
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C
A
mRAT  mRA  mAT
Geometry 10.2 Finding Arc Measures
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Example 1
What have you learned so far?
○
○
○
○
○
○
○
○
○
1) mRS  60
2) mRPS  300
3) 𝑚∠𝑃𝑇𝑄 = 140°
4) mPQR180
5) mQS 100
6) 𝑚∠𝑃𝑇𝑆 = 120°
7) 𝑚∠𝑃𝑇𝑆 = 60°
8) mQSP 220
9) mQTR  40
April 14, 2016
Q
T
60
S
P
Geometry 10.2 Finding Arc Measures
40
R
120
13
Your Turn
A recent survey asked teenagers whether they
would rather meet a famous musician, athlete,
actor, inventor, or other person. The circle graph
shows the results. Find the indicated arc measures.
a.
b.
=220°
=137°
c.
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=223°
d.
=299°
Geometry 10.2 Finding Arc Measures
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Subtending Chords
○ A chord that shares its endpoints with an
arc.
A
O
C
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B
Chord AB
subtends AB.
Chord BC
subtends BC.
Geometry 10.2 Finding Arc Measures
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Congruent Circles Theorem
(Thm 10.3)
Two circles are congruent circles if and
only if they have the same radius.
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Geometry 10.2 Finding Arc Measures
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Congruent Central Angles Theorem
(Thm 10.4)
In the same circle, or in congruent circles,
two minor arcs are congruent if and only if
their corresponding central angles are
congruent.
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Geometry 10.2 Finding Arc Measures
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Example 2
Tell whether the red arcs are congruent.
Explain why or why not.
Congruent, they are
in the same circle
and the central
angles are congruent.
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Not Congruent,
the circles are
not congruent.
Congruent, the circles
have congruent radii so
the circles are congruent
and the central angles
are congruent.
Geometry 10.2 Finding Arc Measures
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Your Turn
Tell whether the red arcs are congruent.
Explain why or why not.
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Geometry 10.2 Finding Arc Measures
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Summary
○ Chords in circles subtend major and
minor arcs.
○ Arcs have the same measure as their
central angles.
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Geometry 10.2 Finding Arc Measures
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Homework
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Geometry 10.2 Finding Arc Measures
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