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Transcript 
Inscribed Angles
• Find measures of inscribed angles
• Find measures of angles of inscribed polygons.
Three congruent central
angles are pictured. What
is the measure of each
angle?
INSCRIBED ANGLES
An inscribed angle is an angle that has its vertex on the
circle and its sides contained in chords of the circle.
B
Vertex B is on
the circle
A
AB and BC are
chords of the circle
ADC is the arc
intercepted by
ABC
D
C
Theorem
Inscribed Angle Theorem
If an angle is inscribed by a circle then the measure of the
angle equals one-half the measure of the intercepted arc
(or the measure of the intercepted arc is twice the
measure of the inscribed angle).
B
A
D
C
1
mABC  (mADC ) or
2
2(mABC)  mADC
Example 1
Measures of Inscribed Angles
In circle O, measure of arc AB = 140,
measure of arc BC = 100, and
measure of arc AD equals the
measure of arc DC.
A
5
O
4
3
D
B
1
5
D
2
O
3
4
C
1
2
Find the measures of the numbered
angles.
A
B
C
Example 1
Measures of Inscribed Angles
In circle O, measure of arc AB = 140,
measure of arc BC = 100, and
measure of arc AD equals the
measure of arc DC.
A
5
O
4
3
D
mAB  mBC  mDC  mAD  360
1
2
Find the measures of the numbered
angles.
First find mDC and mAD
B
C
Theorem
If two inscribed angles of a circle (or congruent circles)
intercept congruent arcs or the same arc, then the angles
are congruent.
A
B
B
A
C
C
D
DAC  DBC
F
E
D
FAE  CBD
Example 2
Inscribed Arcs and Probability
ANGLES OF INSCRIBED POLYGONS
An inscribed triangle with a side that is a diameter is a
special type of triangle.
Theorem
If an inscribed angle intercepts a semicircle,
the angle is a right angle.
B
A
D
C
Example 3
Angles of an Inscribed Triangle
Triangles ABD and ADE are
inscribed in circle F with AB  BD.
Find the measure of each
numbered angle if m1 = 12x – 8
and m2 = 3x + 8.
B
A
3
1
F
4
2
E
D
Example 3
Angles of an Inscribed Triangle
Triangles ABD and ADE are
inscribed in circle F with AB  BD.
Find the measure of each
numbered angle if m1 = 12x – 8
and m2 = 3x + 8.
12 x  8  3 x  8  90
15 x  90
x6
12 x  8  64
3 x  8  26
B
A
3
12x – 8
64°
E
F
4
3x + 8
26°
D
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