Geometry - BakerMath.org
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Transcript Geometry - BakerMath.org
Geometry
Inscribed Angles
Goals
Know what an inscribed angle is.
Find the measure of an inscribed
angle.
Solve problems using inscribed angle
theorems.
July 7, 2015
Inscribed Angle
The vertex is on the
circle and the sides
contain chords of the
circle.
A
B
ABC is an
inscribed angle.
C
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AC is the
intercepted arc.
Inscribed Angle
How does
mABC
compare
to mAC?
A
B
C
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Draw circle O, and points A & B on
the circle. Draw diameter BR.
A
R
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O
B
Draw radius OA and chord AR.
A
2
R
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3
1
O
B
(Very old) Review
The Exterior Angle Theorem (4.2)
The measure of an exterior angle of a
triangle is equal to the sum of the
two remote, interior angles.
2
1
July 7, 2015
m1 + m2 = m3
3
mARO + mOAR = mAOB
A
2
R
3
1
O
What type of
triangle is OAR?
Isosceles
B The base angles
of an isosceles
triangle are
congruent.
1 2
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mARO + mOAR = mAOB
A
• m1 + m2 = m3
• But m1 = m2
2
• m1 + m1 = m3
• 2m1 = m3
R
1
3
O
B
• m1 = (½)m3
This angle is half the
measure of this angle.
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Where we are now.
A
2
R
1
(x/2)
3 x
O
m1 = (½)m3
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x
B
Recall: the measure of
a central angle is
equal to the measure
of the intercepted arc.
m3 m AB
m1 12 m AB
Theorem 12.8
A
x
R
(x/2)
O
B
If an angle is
inscribed in a
circle, then its
measure is onehalf the measure
of the
intercepted arc.
Inscribed Angle Demo
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Example 1
44
?
88
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Example 2
A
mABC ?170
B
85
C
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Example 3
?
60
The circle contains 360.
360 – (100 + 200) = 60
100
30
x
200
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Another Theorem
2x
?
x
x
?
Theorem 10.9
If two inscribed
angles intercept
the same (or
congruent) arcs,
then the angles
are congruent.
Theorem Demonstration
July 7, 2015
A very useful theorem.
Draw a circle.
Draw a diameter.
Draw an inscribed
angle, with the
sides intersecting
the endpoints of
the diameter.
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A very useful theorem.
90
What is the
measure of each
semicircle?
180
What is the
measure of the
inscribed angle?
90
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Theorem 12.10
If an angle is
inscribed in a
semicircle,
then it is a
right angle.
Theorem 12.10 Demo
July 7, 2015
Theorem 12.2: Tangent-Chord
B
C
2
1
A
If a tangent and a
chord intersect at
a point on a circle,
then the measure
of each angle
formed is one-half
the measure of the
intercepted arc.
m1 mAB and m2 mBCA
1
2
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1
2
Simplified Formula
a
b
2 1
m1 a
1
2
m2 b
1
2
7/7/2015
Example 1
80 mAB
1
2
B
C
160
200
80
A
160 mAB
mBCA 360 160
200
Find the mAB and mBCA.
7/7/2015
Example 2. Solve for x.
B
C
8 x 10 x 60
2 x 60
(10x – 60)
4x
A
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4 x 12 (10 x 60)
x 30
Inscribed Polygon
The vertices are all on the same
circle.
The polygon is inside the circle; it is
inscribed.
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July 7, 2015
A
B
D
C
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A cyclic
quadrilateral
has all of its
vertices on
the circle.
An interesting theorem.
B
mBAD 12 mBCD
C
A
D
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An interesting theorem.
mBAD 12 mBCD
B
mBCD
C
A
D
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1
2
mBAD
An interesting theorem.
mBAD 12 mBCD
B
mBCD 12 mBAD
C
A
D
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Adding the equations
together…
An interesting theorem.
mBAD mBCD 12 mBCD 12 mBAD
B
C
A
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D
An interesting theorem.
mBAD mBCD 12 mBCD 12 mBAD
mBAD mBCD
1
2
mBCD mBAD
mBAD mBCD
1
2
360
mBAD mBCD 180
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An interesting theorem.
B
A
mBAD mBCD 180
C
DBAD and BCD are supplementary.
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Theorem 12.11
1
2
4
3
A quadrilateral can
be inscribed in a
circle if and only if
its opposite angles
are supplementary.
Theorem 10.11 Demo
m1 + m3 = 180 & m2 + m4 = 180
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Example
Solve for x and y.
4x + 2x = 180
2x
5y
6x = 180
x= 30
and
5y + 100 = 180
4x
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5y = 80
100
y = 16
Summary
The measure of an inscribed angle is
one-half the measure of the
intercepted arc.
If two angles intercept the same arc,
then the angles are congruent.
The opposite angles of an inscribed
quadrilateral are supplementary.
July 7, 2015
Practice Problems
Inscribed
Hexagon
July 7, 2015