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11.5 Inscribed Angles
Goal
• Use properties of inscribed angles.
Key Vocabulary
•
•
•
•
Inscribed Angle
Intercepted Arc
Inscribed
Circumscribed
Theorems
• 11.7 Inscribed Angle Theorem
• 11.8 Inscribed Triangle
• 11.9 Inscribed Quadrilateral
Inscribed Angle
Inscribed Angle: An angle whose vertex lies on a circle and
whose sides are chords of the circle (or one side tangent to
C
the circle).
ABC is an inscribed angle.
O
Examples:
1
No!
B
A
2
Yes!
D
3
No!
4
Yes!
Intercepted Arc
Intercepted Arc: An angle intercepts an arc if and only if
each of the following conditions holds:
1. The endpoints of the arc lie on the
angle.
2. All points of the arc, except the
endpoints, are in the interior of the angle.
3. Each side of the angle contains an
endpoint of the arc.
C
B
O
ADC is the int ercepted arc of ABC.
A
D
Theorem - Inscribed Angles
• Theorem 11.7 (Inscribed
Angle Theorem):
The measure of an
inscribed angle equals ½
the measure of its
intercepted arc (or the
measure of the
intercepted arc is twice
the measure of the
inscribed angle).
A
C
B
mACB = ½m
2 mACB = m
or
Inscribed Angle Theorem
The measure of an inscribed angle is equal to ½ the measure
of the intercepted arc.
Y
Inscribed Angle
X
55

Z
Intercepted Arc
mYZ
mYXZ 
2
Example 1a
A. Find mX.
Answer: mX = 43
Example 1b
B.
= 2(252) or 104
Your Turn:
A. Find mC.
A. 47
B. 54
C. 94
D. 188
Your Turn:
B.
A. 47
B. 64
C. 94
D. 96
Example 2
In
and
Find the measures of the numbered angles.
Example 2
First determine
Arc Addition
Theorem
20
40
Simplify.
Subtract 168 from
each side.
Divide each side
by 2.
108
Example 2
So,
m
20
40
108
Example 2
20
40
108
Answer:
Your Turn:
A. 30
B. 60
C. 15
D. 120
Your Turn:
A. 110
B. 55
C. 125
D. 27.5
Your Turn:
A. 30
B. 80
C. 40
D. 10
Your Turn:
A. 110
B. 55
C. 125
D. 27.5
Your Turn:
A. 110
B. 55
C. 125
D. 27.5
Comparing Measures of Inscribed
Angles
A
• Find mACB, mADB,
and mAEB.
The measure of each angle is
half the measure of the
intercepted arc.
All three angles intercept the
same arc, arc AB, whose
measure is 60˚.
So the measure of each angle
is 30°.
E
m AB60
B
D
C
Congruent Inscribed Angles
If two inscribed angles of a circle intercept the same
arc or congruent arcs, then the inscribed angles are
congruent.
D
A
A and D both intercept AD,
so, A  D.
P
C
B
Example 3
Find mR.
R  S
mR  mS
12x – 13 = 9x + 2
x =5
R and S both intercept
.
Definition of congruent angles
Substitution
Simplify.
Answer: So, mR = 12(5) – 13 or 47.
Your Turn:
Find mI.
A. 4
B. 25
C. 41
D. 49
Example 4: Find the value of
x and y in the figures
F
y
A
A
40˚
D
B
B
50˚
50˚
y
x
x
C
E
m AC
50 
2
m AC  100
m AC 100
xy

 50
2
2
C
E
mAD 40
x

 20
2
2
mAD  mDC 40  y
50 

2
2
100  40  y
y  60
Polygons
Inscribed Polygon:
A polygon inside the circle whose vertices
lie on the circle.
Circumscribed Polygon :
A polygon whose sides are
tangent to a circle.
Angles of Inscribed Polygons
• Theorem 11.8:
An inscribed angle of a
triangle intercepts a
diameter or semicircle
if and only if the angle
is a right angle.
i.e. If AC is a diameter
of ⊙O, then the
mABC = 90°
o
Example 5
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
Angle Sum Theorem
Substitution
Simplify.
9x = 90
Subtract 90 from each
side.
x = 10
Divide each side by 9.
Answer: So, mB = 8(10) – 4 or 76.
Your Turn:
Find mD.
A. 8
B. 16
C. 22
D. 28
Ex. 6:
Lesson 4 Ex4
Your Turn:
A. 45
B. 90
C. 180
D. 80
Your Turn:
A. 17
B. 76
C. 60
D. 42
Your Turn:
A. 17
B. 76
C. 60
D. 42
Your Turn:
A. 73
B. 30
C. 60
D. 48
Angles of Inscribed Polygons
• Theorem 11.9:
If a quadrilateral is
inscribed in a ⊙, then its
opposite s are
supplementary.
i.e. Quadrilateral ABCD
is inscribed in ⊙O, thus
A and C are
supplementary and B
and D are
supplementary.
A
D
B
O
C
Theorem 11.9
If a quadrilateral is inscribed
in a circle, then its opposite
angles are supplementary.
Example 7:
Quadrilateral QRST is inscribed in
find
and
Draw a sketch of this situation.
Answer:
If
and
Your Turn:
Quadrilateral BCDE is inscribed in
find
and
Answer:
If
and
Example 8
An insignia is an emblem that signifies rank,
achievement, membership, and so on. The
insignia shown is a quadrilateral inscribed in a
circle. Find mS and mT.
Example 8
Since TSUV is inscribed in a circle, opposite angles are
supplementary.
S + V = 180
S + V = 180
S + 90 = 180
(14x) + (8x + 4) = 180
S = 90
22x + 4 = 180
22x = 176
x =8
Answer: So, mS = 90 and mT = 8(8) + 4 or 68.
Your Turn:
An insignia is an emblem that signifies rank,
achievement, membership, and so on. The insignia
shown is a quadrilateral inscribed in a circle. Find
mN.
A. 48
B. 36
C. 32
D. 28
REVIEW PROBLEMS
Example 1
Find Measures of Inscribed Angles and Arcs
Find the measure of the inscribed angle or the
intercepted arc.
a.
b.
SOLUTION
a. mNMP = 1 mNP
2
1
= (100°)
2
= 50°
The measure of an inscribed angle is
half the measure of its intercepted arc.
Substitute 100° for mNP.
Simplify.
Example 1
Find Measures of Inscribed Angles and Arcs
b.
b.
mZYX =
1
mZWX
2
105° =
1
mZWX
2
210° = mZWX
The measure of an inscribed angle is
half the measure of its intercepted arc.
Substitute 105° for mZYX.
Multiply each side by 2.
Your Turn:
Find the measure of the inscribed angle or the
intercepted arc.
1.
ANSWER
mBAC = 45°
ANSWER
mDEF = 80°
ANSWER
mKNP = 240°
2.
3.
Example 2
Find Angle Measures
Find the values of x and y.
SOLUTION
Because ∆ABC is inscribed in a circle and AB is a
diameter, it follows from Theorem 11.8 that ∆ABC is a
right triangle with hypotenuse AB.
Therefore, x = 90. Because A and B are acute angles
of a right triangle, y = 90 – 50 = 40.
Your Turn:
Find the values of x and y in C.
4.
ANSWER
x = 90; y = 55
ANSWER
x = 45; y = 90
ANSWER
x = 30; y = 90
5.
6.
Example 3
Find Angle Measures
Find the values of y and z.
SOLUTION
Because RSTU is inscribed in a circle, by Theorem 11.9
opposite angles must be supplementary.
S and U are
opposite angles.
R and T are
opposite angles.
mS + mU = 180°
mR + mT = 180°
120° + y° = 180°
y = 60
z° + 80° = 180°
z = 100
Your Turn:
Find the values of x and y in C.
7.
ANSWER
x = 85; y = 80
ANSWER
x = 90; y = 90
ANSWER
x = 130; y = 100
8.
9.
66˚
54˚
43˚
90˚
54˚
50˚
54˚
36˚
50˚
72˚
180˚
In the figure, find the value of x.
X = 102
Find the value of x.
X = 21.5
Assignment
• Pg. 617 – 619; #1 – 29 odd, 33 – 49 odd