8-5 Angles in Circles

1
8-5 Angles in Circles
2
Central Angles

• A central angle is an angle whose vertex is the
  CENTER of the circle

NOT A Central Angle (of a circle)
Central Angle (of a circle)
Central Angle (of a circle)
3
CENTRAL ANGLES AND ARCS

• The measure of a central angle is equal to the
  measure of the intercepted arc.

4
CENTRAL ANGLES AND ARCS

• The measure of a central angle is equal to the
  measure of the intercepted arc.

Central Angle
Intercepted Arc
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EXAMPLE

• Segment AD is a diameter. Find the values of x
  and y and z in the figure.
• x 25
• y 100
• z 55

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SUM OF CENTRAL ANGLES
The sum of the measures fo the central angles of
a circle with no interior points in common is
360º.
360º
7
Find the measure of each arc.
D
C
2x-14
4x
2x
3x
E
B
3x10
4x 3x 3x 10 2x 2x 14 360 x
26 104, 78, 88, 52, 66 degrees
A
8
Inscribed Angles
An inscribed angle is an angle whose vertex is on
a circle and whose sides contain chords.
3
2
1
4
Is NOT!
Is SO!
Is NOT!
Is SO!
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INSCRIBED ANGLE THEOREM
Thrm 9-7. The measure of an inscribed angle is
equal to ½ the measure of the intercepted arc.
The measure of an inscribed angle is equal to ½
the measure of the intercepted arc.
10
INSCRIBED ANGLE THEOREM
Thrm 9-7. The measure of an inscribed angle is
equal to ½ the measure of the intercepted arc.
The measure of an inscribed angle is equal to ½
the measure of the intercepted arc.
11
INSCRIBED ANGLE THEOREM
Thrm 9-7. The measure of an inscribed angle is
equal to ½ the measure of the intercepted arc.
The measure of an inscribed angle is equal to ½
the measure of the intercepted arc.
Inscribed Angle
Y
110?
55?
Z
Intercepted Arc
12
Thrm 9-7. The measure of an inscribed angle is
equal to ½ the measure of the intercepted arc.
Find the value of x and y in the figure.

• X 20
• Y 60

P
40?
Q
50?
y?
S
x?
R
T
13
Corollary 1. If two inscribed angles intercept
the same arc, then the angles are congruent..
Find the value of x and y in the figure.

• X 50
• Y 50

P
Q
y?
50?
S
R
x?
T
14
An angle formed by a chord and a tangent can be
considered an inscribed angle.
15
An angle formed by a chord and a tangent can be
considered an inscribed angle.
P
Q
S
R
m?PRQ ½ mPR
16
What is m?PRQ ?
P
Q
60?
S
R
17
An angle inscribed in a semicircle is a right
angle.
P
180?
R
18
An angle inscribed in a semicircle is a right
angle.
P
180?
90?
S
R
19
Interior Angles

• Angles that are formed by two intersecting
  chords. (Vertex IN the circle)

A
D
B
C
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Interior Angle Theorem

• The measure of the angle formed by the two chords
  is equal to ½ the sum of the measures of the
  intercepted arcs.

21
Interior Angle Theorem

• The measure of the angle formed by the two chords
  is equal to ½ the sum of the measures of the
  intercepted arcs.

22
Interior Angle Theorem
91?
A
C
x
y
B
D
85?
23
Exterior Angles

• An angle formed by two secants, two tangents, or
  a secant and a tangent drawn from a point outside
  the circle. (vertex OUT of the circle.)

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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.

j?
k?
k?
1
j?
j?
k?
1
1
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Exterior Angle Theorem

• The measure of the angle formed is equal to ½ the
  difference of the intercepted arcs.

26
Find

• ltC ½(265-95)
• ltC ½(170)
• mltC 85