Download Inscribed Angles

Geometry
Inscribed Angles
Goals
 Know what an inscribed angle is.
 Find the measure of an inscribed
angle.
 Solve problems using inscribed angle
theorems.
April 30, 2017
Inscribed Angle
The vertex is on the
circle and the sides
contain chords of the
circle.
A
B
ABC is an
inscribed angle.
C
April 30, 2017
AC is the
intercepted arc.
Inscribed Angle
How does
mABC
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
April 30, 2017
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
April 30, 2017
m1 + m2 = m3
3
mARO + mOAR = mAOB
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
April 30, 2017
mARO + mOAR = mAOB
A
• m1 + m2 = m3
• But m1 = m2
2
• m1 + m1 = m3
• 2m1 = m3
R
1
3
O
B
• m1 = (½)m3
This angle is half the
measure of this angle.
April 30, 2017
Where we are now.
A
2
R
1
(x/2)
3 x
O
m1 = (½)m3
April 30, 2017
x
B
Recall: the measure of
a central angle is
equal to the measure
of the intercepted arc.
m3  m AB
m1  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
April 30, 2017
Example 1
44
?
88
April 30, 2017
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
April 30, 2017
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
April 30, 2017
A very useful theorem.
Draw a circle.
Draw a diameter.
Draw an inscribed
angle, with the
sides intersecting
the endpoints of
the diameter.
April 30, 2017
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
April 30, 2017
Theorem 12.2: Tangent-Chord
B
C
2
A
1
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.
m1  12 mAB and m2  12 m BCA
4/30/2017
Simplified Formula
a
b
2 1
m1  a
1
2
m2  b
1
2
4/30/2017
Example 1
80  m AB
1
2
B
C
160
200
80
A
160  m AB
mBCA  360  160
 200
Find the mAB and mBCA.
4/30/2017
Example 2. Solve for x.
B
C
(10x – 60)
4x
A
4/30/2017
4 x  12 (10 x  60)
8 x  10 x  60
2 x  60
x  30
Inscribed Polygon
 The vertices are all on the same
circle.
 The polygon is inside the circle; it is
inscribed.
April 30, 2017
April 30, 2017
A
B
D
C
April 30, 2017
A cyclic
quadrilateral
has all of its
vertices on
the circle.
An interesting theorem.
B
mBAD  mBCD
1
2
C
A
D
April 30, 2017
An interesting theorem.
mBAD  mBCD
B
1
2
mBCD  mBAD
1
2
C
A
D
April 30, 2017
An interesting theorem.
mBAD  12 mBCD
B
mBCD  12 mBAD
C
A
D
April 30, 2017
Adding the equations
together…
An interesting theorem.
mBAD  mBCD  12 mBCD  12 mBAD
B
C
A
April 30, 2017
D
An interesting theorem.
mBAD  mBCD  12 mBCD  12 mBAD
mBAD  mBCD 
1
2
 mBCD  mBAD 
mBAD  mBCD 
1
2
 360 
mBAD  mBCD  180
April 30, 2017
An interesting theorem.
B
A
mBAD  mBCD  180
C
DBAD and BCD are supplementary.
April 30, 2017
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
m1 + m3 = 180 & m2 + m4 = 180
April 30, 2017
Example
Solve for x and y.
4x + 2x = 180
2x
5y
6x = 180
x= 30
and
5y + 100 = 180
4x
April 30, 2017
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.
April 30, 2017
Practice Problems
Inscribed
Hexagon
April 30, 2017