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11-311-3Inscribed AnglesInscribed Angles
Objective:Objective: To find the To find the measure of an inscribed angle.measure of an inscribed angle.
Central Angle
(of a circle)
Central Angle
(of a circle)
NOT A Central Angle
(of a circle)
Central AngleAn angle whose vertex lies on the center of the circle.Definition:
Central Angle TheoremThe measure of a center angle is equal to the measure of the
intercepted arc.
AD
Y
Z
O 110
110
Intercepted Arc Center Angle
Example:
Give is the diameter, find the value of x and y and z in the figure.
z
25
55y
x
O
B
D
AC 25
180 (25 55 ) 180 80 100
55
x
y
z
Vocabulary
Inscribed Angle
Intercepted Arc
B
A
C
angle inscribedan is CCAB of arc dintercepte theis
Theorem 11-9 (Inscribed Angle Theorem) The measure of an inscribed angle is half the
measure of its intercepted arc.
B
C
A
AC2
1B mm
Example 1: Using the Inscribed Angle Theorem
P
Q
R
S
T
bo
ao
30o
60o
60 if PQR Find mRSm
60o
902
1 TQR m
54 TQR m
6054 PQR m
051 PQR m
Example 2: Find the value of x and y in the figure.
4020
2 2
4050
2 2
100 40 60
mAD
mAD mD
y y
C y
x
y
40
x
50
A
B
C
D
E
Corollaries to the Inscribed Angle Theorem
1. Two inscribed angles that intercept the same arc are congruent.
2. An angle inscribed in a semicircle is a right angle.
3. The opposite angles of a quadrilateral inscribed in a circle are supplementary.
An angle inscribed in a semicircle is a right angle.
R
P 180
S90
Example 3: Using Corollaries to Find Angle Theorem
60o
80o
1
2
3
4
Find the diagram at the right, find the measure of each numbered angle.
90 1m
1402
1 4 m 70
120o
100o
093 m
2202
1 2 m 110
Example 4: Find the value of Example 4: Find the value of x and y.x and y.
85 + x = 180
x = 95
80 + y = 180
y = 100xo
yo
80o
85o
Theorem 11-10
BDC2
1C mm
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
B
C
D
BD
C
Example 5: Using Theorem 11-10
35o
yo
xo
Q
L
K
J
Find x and y. 90 QJL m90o
55y
125180 y
JL2
1 xm
(70)2
1 x m
35xm 35x
Assignment
Page 601
#1-23
