11-3 Inscribed Angles

Transcript 11-3 Inscribed Angles

11-3
Inscribed Angles
Objective: To find the
measure of an inscribed
angle.
Central Angle
Definition: An angle whose vertex lies on the center of the circle.
Central
Angle
(of a circle)
Central
Angle
(of a circle)
NOT A
Central
Angle
(of a circle)
Central Angle Theorem
The measure of a center angle is equal to the measure of the
intercepted arc.
Y
Center Angle
Intercepted Arc
Give AD is the diameter, find the
value of x and y and z in the figure.
O
110
Example:
B
25
A
C
x
y  180  (25  55 )  180  80  100
y
O
x  25
Z
55
z
D
z  55
Vocabulary
A
Intercepted Arc
Inscribed Angle
C
B
AB is theintercepted arc of C
C is an inscribed angle
Theorem 11-9 (Inscribed Angle Theorem)

The measure of an inscribed angle is half the
measure of its intercepted arc.
A
1
mB  mAC
2
B
C
Example 1: Using the Inscribed Angle
Theorem

Find mPQR if mRS  60
P
1
mTQR  90 
2
mT QR  45
mPQR  45  60
Q
ao
60o
T
30o
S
mPQR  105
bo
R
60o
Example 2: Find the value of x and y
A
in the figure.
40 °
D
B
50 °
y°
x°
C
E
m AD 40
x

 20
2
2
m AD  mDC 40  y
50 

2
2
100  40  y
y  60
Corollaries to the Inscribed Angle Theorem
1.
2.
3.
Two inscribed angles that intercept the same arc
are congruent.
An angle inscribed in a semicircle is a right
angle.
The opposite angles of a quadrilateral inscribed
in a circle are supplementary.
An angle inscribed in a semicircle is a
right angle.
P
S
180
90
R
Example 3: Using Corollaries to Find Angle
Theorem

Find the diagram at the
right, find the measure of
each numbered angle.
m1  90
1
m4  140   70
2
m3  90
1
m2  220   110
2
120o
1
60o
2
4
80o
3
100o
Example 4: Find the value of x and y.
xo
yo
85o
80o
85 + x = 180
x = 95
80 + y = 180
y = 100
Theorem 11-10

The measure of an angle formed by a
tangent and a chord is half the measure of
the intercepted arc.
1
mC  mBDC
2
B
B
D
D
C
C
Example 5: Using Theorem 11-10

mQJL  90
Find x and y.
J
y  180 125
y  55
1
mx  JL
2
1
mJQL  JL
2
mx  mJQL
90o
Q
35o
xo
yo
L
K
x  35
Assignment
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