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```10-4 Inscribed Angles
You found measures of interior angles of
polygons.
• Find measures of inscribed angles.
• Find measures of angles of inscribed
polygons.
Find the measure of each arc
• mDE
• mAED
• mEBD
54°
E
180°
54°
A
D
306°
C
B
What kind of angle is angle ECD?
Central angle
Definition
His name was inscribed on the award.
The square is inscribed in the circle.
B
An inscribed angle is an
angle with its vertex on a
circle and sides that
contain chords of the circle.
C
A
Intercepted Arc
An intercepted arc has endpoints
on the sides of an inscribed angle
and lies in the interior of the
inscribed angle
R
C
Q
Intercepted arc
P
Center P is on a side
of the inscribed angle
P
Center P is inside
the inscribed
angle
S
P
Center P is in the
exterior of the
inscribed angle
Sizing Up Inscribed Angles
A
• Measure the central
angle ACB
• What is the measure of
the arc AB?
• Measure the inscribed
• Compare the measure
of the central angle and
the inscribed angle.
C
D
B
The measure of an inscribed angle is half
the measure of its intercepted arc.
A
C
B
p. 723
p. 724
Find the measure of each angle or
arc indicated by a variable.
x°
C
24°
z°
160°
48°
y°
80°
90°
A. Find mX.
B.
= 2(52) or 104
A. Find mC.
A. 47
B. 54
C. 94
D. 188
A
D
If two inscribed angles
intercept the same arc,
then they are congruent.
B
C
p. 724
ALGEBRA Find mR.
R  S
R and S both intercept
.
mR  mS
Definition of congruent angles
12x – 13 = 9x + 2
Substitution
x =5
Simplify.
Answer: So, mR = 12(5) – 13 or 47.
ALGEBRA Find mI.
A. 4
B. 25
C. 41
D. 49
An inscribed angle that
intercepts a semicircle is
a right angle.
p. 725
ALGEBRA Find mB.
ΔABC is a right triangle because
C inscribes a semicircle.
mA + mB + mC = 180
(x + 4) + (8x – 4) + 90 = 180
9x + 90 = 180
9x = 90
Angle Sum Theorem
Substitution
Simplify.
Subtract 90 from each
side.
x = 10
Divide each side by 9.
Answer: So, mB = 8(10) – 4 or 76.
10-4 Assignment
Page 727, 11-18, 23-24
```
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