Transcript 9-5 Inscribed Angles

Lesson 8-5
Angle
Formulas
Lesson 8-5: Angle Formulas
1
Central Angle
Definition: An angle whose vertex lies on the center of the circle.
Central
Angle
(of a circle)
Central
Angle
(of a circle)
Lesson 8-5: Angle Formulas
NOT A
Central
Angle
(of a circle)
2
Central Angle Theorem
The measure of a center angle is equal to the measure of the
intercepted arc.
Y
Intercepted Arc
Center Angle
Example: Give AD is the diameter, find the
value of x and y and z in the figure.
O
110
B
25
A
C
x
y
O
55
z
D
Z
x  25
y  180  (25  55 )  180  80  100
z  55
Lesson 8-5: Angle Formulas
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Example: Find the measure of each arc.
4x + 3x + (3x +10) + 2x + (2x-14) =
B 360°14x – 4 = 360°
14x = 364°
x = 26°
2x-14 C
4x
E
4x = 4(26) = 104°
2x
3x
3x = 3(26) = 78°
3x+10
A
D
3x +10 = 3(26) +10= 88°
2x = 2(26) = 52°
2x – 14 = 2(26) – 14 = 38°
Lesson 8-5: Angle Formulas
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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 the circle).
ABC is an inscribed angle.
No!
B
O
Examples:
1
C
A
D
3
2
Yes!
No!
Lesson 8-5: Angle Formulas
4
Yes!
5
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
ADC is the int ercepted arc of ABC.
O
A
Lesson 8-5: Angle Formulas
D
6
Inscribed Angle Theorem
The measure of an inscribed angle is equal to ½ the measure of the
intercepted arc.
Y
Inscribed Angle
A
55
C
D
Z
Intercepted Arc
B
An angle formed by a chord and a tangent can be considered an
inscribed angle.
mAB
mABC 
2
Lesson 8-5: Angle Formulas
7
Examples: Find the value of x and y in the fig.
F
y°
A
A
40°
D
B
B
50°
50°
y°
x°
x°
C
C
E
m AC
50 
2
m AC  100
m AC 100
x y

 50
2
2
m AD 40 E
x

 20
2
2
m AD  mDC 40  y
50 

2
2
100  40  y
y  60
Lesson 8-5: Angle Formulas
8
An angle inscribed in a semicircle is a
right angle.
P
S
180
90
R
Lesson 8-5: Angle Formulas
9
Interior Angle Theorem
Definition: Angles that are formed by two intersecting chords.
A
D
AEC and DEB are int erior angles.
2
1
E
B
Interior Angle Theorem:
C
The measure of the angle formed by the two intersecting chords is
equal to ½ the sum of the measures of the intercepted arcs.
mAC  mDB
m1  m2 
2
Lesson 8-5: Angle Formulas
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Example: Interior Angle Theorem
A
91
C
x
y °
°
x  88
B
D
85
1
x  ( m AC  mDB )
2
1
x  (91  85 )
2
y  180  88
y  92
Lesson 8-5: Angle Formulas
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Exterior Angles
An angle formed by two secants, two tangents, or a secant and a
tangent drawn from a point outside the circle.
x°
y°1 x°
y°2
x°
Two secants
y° 3
2 tangents
A secant and a tangent
Lesson 8-5: Angle Formulas
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Exterior Angle Theorem
The measure of the angle formed is equal to ½ the
difference of the intercepted arcs.
x
y 1
x y
m1 
2
x
y 2
x
x y
m2 
2
Lesson 8-5: Angle Formulas
y 3
x y
m3 
2
13
Example: Exterior Angle Theorem
In the given figure find the mACB.
B
265
95
A
1
mACB  (mADB  mAD)
2
1
mACB  (265  95 )
2
C
1
mACB  (170 )  85
2
Lesson 8-5: Angle Formulas
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Given AF is a diameter , mAG  100 , mCE  30 and mEF  25 .
Find the measure of all numbered angles.
m1  mFG  80
Dm2  m AG  100
C
A
3
100°
Q
2 1
6
1
55
m3  (mCE  mEF ) 
 22.5
30°
2
2
E
1
80  155
m

4

(
mGF

m
ACE
)

 117.5
25°
5
2
2
4 F m5  180  117.5  62.5
1
1
m6  (m AG  mCE )  (100  30)  35
2
2
G
Lesson 8-5: Angle Formulas
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Inscribed Quadrilaterals
If a quadrilateral is inscribed in a circle, then the opposite
angles are supplementary.
A
B
mDAB + mDCB = 180 
mADC + mABC = 180 
D
C
Lesson 8-5: Angle Formulas
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