geometry inscribed angles august 24, 2015 goals know what an inscribed angle is. find the measure...
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Geometry
Inscribed Angles
April 19, 2023
Goals
Know what an inscribed angle is. Find the measure of an inscribed
angle. Solve problems using inscribed angle
theorems.
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Inscribed Angle
The vertex is on the circle and the sides contain chords of the circle.
A
C
B ABC is an inscribed angle.
AC is the intercepted arc.
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Inscribed Angle
A
C
B
How does mABC compare to mAC?
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Draw circle O, and points A & B on the circle. Draw diameter BR.
OB
A
R
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Draw radius OA and chord AR.
OB
A
R 1
2
3
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(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.
1
2
3
m1 + m2 = m3
April 19, 2023
mARO + mOAR = mAOB
OB
A
R
What type of triangle is OAR?
Isosceles
The base angles of an isosceles triangle are congruent.
1 2
1
2
3
April 19, 2023
mARO + mOAR = mAOB
OB
A
R
• m1 + m2 = m3
• But m1 = m2
• m1 + m1 = m3
• 2m1 = m3
• m1 = (½)m3
This angle is half the measure of this angle.
1
2
3
April 19, 2023
Where we are now.
OB
A
R x(x/2)
Recall: the measure of a central angle is equal to the measure of the intercepted arc.
12
3
1
m mAB
m mAB
x
m1 = (½)m3
1
2
3
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Theorem 12.8
OB
A
R (x/2)
If an angle is inscribed in a circle, then its measure is one-half the measure of the intercepted arc.
x
Inscribed Angle Demo
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Example 1
88
?44
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Example 2
A
B
C
85
mABC ?170
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Example 3
x
200
100
The circle contains 360.
360 – (100 + 200) = 60
30
?60
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Another Theorem
2x
xx
?
?
Theorem 10.9
If two inscribed angles intercept the same (or congruent) arcs, then the angles are congruent.Theorem Demonstration
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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.
What is the measure of each semicircle?
180
What is the measure of the inscribed angle?
90
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
04/19/23
Theorem 12.2: Tangent-Chord
A
BC
12
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.
1 122 and1 2m mA m m CAB B
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Simplified Formula
ab
12
12
12
1
2
m a
m b
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Example 1
Find the and .mAB mBCA
1280
160
mAB
mAB
A
BC
80 360 160
200
mBCA
160200
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Example 2. Solve for x.
A
BC
4x
(10x – 60)
124 (10 60)
8 10 60
2 60
30
x x
x x
x
x
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Inscribed Polygon
The vertices are all on the same circle.
The polygon is inside the circle; it is inscribed.
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April 19, 2023
A cyclic quadrilateral has all of its vertices on the circle.
B
A
C
D
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An interesting theorem.
m BAD
A
B
C
D
12mBCD
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An interesting theorem.
m BAD 12mBCD
m BCD
A
B
C
D
12mBAD
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An interesting theorem.
1
12
2
m BC
m BAD
D mB D
mBCD
A
A
B
C
D
Adding the equations together…
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An interesting theorem.
A
B
C
D
1 12 2m BAD m BCD mBCD mBAD
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An interesting theorem.
1 12 2
12
12 360
180
m BAD m BCD mBCD mBAD
m BAD m BCD mBCD mBAD
m BAD m BCD
m BAD m BCD
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An interesting theorem.
A
B
C
DBAD and BCD are supplementary.
180m BAD m BCD
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Theorem 12.11
1
3
4
2
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
m1 + m3 = 180 & m2 + m4 = 180
Theorem 10.11 Demo
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Example Solve for x and y.
4x
2x5y
100
4x + 2x = 180
6x = 180
x= 30
and
5y + 100 = 180
5y = 80
y = 16
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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.
April 19, 2023
Practice Problems
Inscribed
Hexagon
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