Download of a circle

Circles
Part 2
Geometry
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: Given 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
3
Example: Find the measure of each arc.
4x + 3x + (3x +10) + 2x + (2x-14) = 360°
14x – 4 = 360°
B
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
4
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 intercepted arc of ABC
O
A
Lesson 8-5: Angle Formulas
D
5
Intercepted Arc
• Intercepted arcs
for 1
K
M
– KM and KL
1
L
2
• Intercepted arcs
for 2
O
N
– NO and NPO
P
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!
7
Using Inscribed Angles
• An inscribed angle is
an angle whose
vertex is on a circle
and whose sides
contain chords of
the circle. The arcinscribed angle
that lies in the
interior of an
inscribed angle and
has endpoints on
the angle is called
the intercepted arc
of the angle.
intercepted arc
Theorem: Measure of an Inscribed Angle
A
• If an angle is
inscribed in a
circle, then its
measure is one
half the measure
of its intercepted
arc.
mADB = ½m AB

C
D
B
Ex. : Finding Measures of Arcs
and Inscribed Angles
• Find the measure
of the blue arc or
angle.
S
R

m QTS = 2mQRS =
T
2(90°) = 180°
Q
Ex. : Finding Measures of Arcs
and Inscribed Angles
• Find the measure
of the blue arc.

m ZWX = 2mZYX =
2(115°) = 230°
W
Z
115
Y
X
Ex. : Finding Measures of Arcs
and Inscribed Angles
• Find the measure
of the blue arc or
angle.


m NMP = ½ m NP
N
100°
M
P
½ (100°) = 50°
Corollary
• If a right triangle is inscribed in a circle, then the
hypotenuse is a diameter of the circle.
Conversely, if one side of an inscribed triangle is a
diameter of the circle, then the triangle is a right
triangle and the angle opposite the diameter is the
right angle.
• B is a right angle if and only if AC is a diameter
of the circle.
A
O
B
C
Using Corollary
B
• Find the value of x.
• AB is a diameter.
So, C is a right
angle and mC =
90°
• 2x° = 90°
• x = 45
Q
A
2x°
C
Corollary:
A
• If two inscribed
angles of a circle
intercept the
same arc, then
the angles are
congruent.
• C  D
D
B
C
Ex. : Comparing Measures of
Inscribed Angles
A
• Find mACB,
mADB, and
mAEB.
60
E
The measure of each
angle is half the
measure of AB
m AB = 60°, so the
measure of each
angle is 30°

B

D
C
Ex. : Finding the Measure of an
Angle
G
• It is given that
mE = 75°. What
is mF?

• E and F both
intercept GH , so
E  F. So,
mF = mE = 75°
E
75°
F
H
Ex. : Using the Measure of an
Inscribed Angle
• Theater Design.
When you go to the
movies, you want to
be close to the
movie screen, but
you don’t want to
have to move your
eyes too much to
see the edges of the
picture.
Ex. : Using the Measure of an
Inscribed Angle
• If E and G are the
ends of the
screen and you
are at F, mEFG
is called your
viewing angle.
Ex. : Using the Measure of an
Inscribed Angle
• You decide that
the middle of the
sixth row has the
best viewing
angle. If
someone else is
sitting there,
where else can
you sit to have
the same viewing
angle?
Ex. : Using the Measure of an
Inscribed Angle
• Solution: Draw
the circle that is
determined by
the endpoints of
the screen and
the sixth row
center seat. Any
other location on
the circle will
have the same
viewing angle.
Theorem
• If a tangent and a
secant, two tangents
or two secants
intercept in the
EXTERIOR of a circle,
then the measure of
the angle formed is
one half the
difference of the
measures of the
intercepted arcs.
B
A
1
C

m1 = ½ m( BC - AC )
Theorem
• If a tangent and a
secant, two tangents
or two secants
intercept in the
EXTERIOR of a circle,
then the measure of
the angle formed is
one half the
difference of the
measures of the
intercepted arcs.
P
2
Q
R
m2 = ½

m(PQR
- PR )
Theorem
X
W
• If a tangent and a
secant, two tangents 3
Z
or two secants
intercept in the
EXTERIOR of a circle,
Y
then the measure of
the angle formed is
one half the
difference of the
measures of the
m3 = ½ m( XY - WZ )
intercepted arcs.

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 y
m2 
2
x
y 3
x y
m3 
2
E
Ex. : Using Theorem
200°
• Find the value of x
Solution:
D
F
x°
 
mGHF = ½ m(EDG
G
- GF ) Apply Theorem
72° = ½ (200° - x°)
144 = 200 - x°
- 56 = -x
56 = x
H
72°
Substitute values.
Multiply each side by 2.
Subtract 200 from both sides.
Divide by -1 to eliminate
negatives.
Ex. : Using Theorem
 
M
Because MN and MLN make a
whole circle, m MLN =360°-92°=268°
L
• Find the value of x
Solution:
 
mGHF = ½ m(MLN - m MN )
= ½ (268 - 92)
= ½ (176)
= 88
92° x°
N
Apply Theorem 10.14
Substitute values.
Subtract
Multiply
P
D
Theorem
• If two chords intersect
in the interior of a
circle, then the
measure of each angle
is one half the sum of
the measures of the
arcs intercepted by the
angle and its vertical
angle.
1
A
2
B
 
 
m1 = ½ m CD + m AB
m2 = ½ m BC + m AD
C
Ex. : Finding the Measure of an
Angle Formed by Two Chords
P
106°
• Find the value of x
S
Q
x°
R
 
• Solution:
x° = ½ (mPS +m RQ )
x° = ½ (106° + 174°)
x = 140
174°
Apply Theorem
Substitute values
Simplify