10-4 Inscribed Angles
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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.
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Find the measure of each arc
• mDE
• mAED
• mEBDA
B
C
D
E
54°
54°
180°
306°
What kind of angle is angle ECD?
Central angle
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DefinitionHis name was inscribed on the award.
The square is inscribed in the circle.
A
B
CAn inscribed angle is an angle with its vertex on a circle and sides that contain chords of the circle.
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Intercepted ArcAn intercepted arc has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle
Q
S
C
R
Intercepted arc
P P P
Center P is on a side of the inscribed angle
Center P is inside the inscribed angle
Center P is in the exterior of the inscribed angle
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Sizing Up Inscribed Angles
• Measure the central angle
• What is the measure of the arc AB?
• Measure the inscribed angle
• Compare the measure of the central angle and the inscribed angle.
A
BC
D
ACB
ADB
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p. 723
The measure of an inscribed angle is half the measure of its intercepted arc.
A
B
C
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p. 724
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Find the measure of each angle or arc indicated by a variable.
24°
x°
160°
y°
C
z°
48°
80°
90°
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A. Find mX.
Answer: mX = 43
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B.
= 2(52) or 104
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A. 47
B. 54
C. 94
D. 188
A. Find mC.
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If two inscribed angles intercept the same arc, then they are congruent.
A
B
C
D
p. 724
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ALGEBRA Find mR.
R S R and S both intercept . mR mS Definition of congruent angles
12x – 13 = 9x + 2 Substitutionx = 5 Simplify.
Answer: So, mR = 12(5) – 13 or 47.
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A. 4
B. 25
C. 41
D. 49
ALGEBRA Find mI.
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An inscribed angle that intercepts a semicircle is a right angle.
p. 725
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ALGEBRA Find mB.
ΔABC is a right triangle because C inscribes a semicircle.
mA + mB + mC = 180 Angle Sum Theorem(x + 4) + (8x – 4) + 90 = 180 Substitution
9x + 90 = 180 Simplify.9x = 90 Subtract 90 from each
side.x = 10 Divide each side by 9.
Answer: So, mB = 8(10) – 4 or 76.
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10-4 Assignment
Page 727, 11-18, 23-24