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8-5 Angles in Circles
Central Angles
Central Angle
(of a circle)
Central Angle
(of a circle)
NOT A Central Angle
(of a circle)
• A central angle is an angle whose vertex is the CENTER of the circle
CENTRAL ANGLES AND ARCS
The measure of a central angle is equal to the measure of the intercepted arc.
CENTRAL ANGLES AND ARCS
The measure of a central angle is equal to the measure of the intercepted arc.
Y
Z
O 110110
Intercepted Arc
Central Angle
EXAMPLE
• Segment AD is a diameter. Find the values of x and y and z in the figure.
x = 25°
y = 100°
z = 55°
A
B
O
C
D
55x y
25
z
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
Find the measure of each arc.
A
E B
C
D
2x
2x-1
4
4x
3x 3x+10
4x + 3x + 3x + 10+ 2x + 2x – 14 = 360…x = 26104, 78, 88, 52, 66 degrees
An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords.
Inscribed Angles
1 423
Is NOT!
Is NOT!
Is SO! Is SO!
Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.INSCRIBED ANGLE THEOREM
The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.
x
x
Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.INSCRIBED ANGLE THEOREM
The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.
1
2
Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.INSCRIBED ANGLE THEOREM
The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.
Y
Z
55110
Inscribed Angle
Intercepted Arc
Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.
x
y
Q
R
P
S
T
50
40
Find the value of x and y in the figure.
• X = 20°
• Y = 60°
Corollary 1. If two inscribed angles intercept the same arc, then the angles are congruent..
x R
Q
S
T
50
Py
Find the value of x and y in the figure.
• X = 50°
• Y = 50°
An angle formed by a chord and a tangent can be considered an inscribed angle.
2x
An angle formed by a chord and a tangent can be considered an inscribed angle.
R
S
P
Q
mPRQ = ½ mPR
What is mPRQ ?
R
S
P
Q
60
An angle inscribed in a semicircle is a right angle.
R
P 180
An angle inscribed in a semicircle is a right angle.
R
P 180
S90
• Angles that are formed by two intersecting chords. (Vertex IN the circle)
Interior Angles
A
B
C
D
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.
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.
1
A
B
C
D 1m 1 (mAC mBD)
2
A
B
C
D
x°
91
85
Interior Angle Theorem
91 5(2
81
)x
88xy°
88180 y
92y
• 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.)
Exterior Angles
• An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle.
Exterior Angles
1 jk 1jk1jk
Exterior Angle Theorem
• The measure of the angle formed is equal to ½ the difference of the intercepted arcs.
1jk 3jk1jk
1m 1 (k j)
2
Find m ACB
• <C = ½(265-95)
• <C = ½(170)
• m<C = 85°
265
95C
B
A
PUTTING IT TOGETHER!
• AF is a diameter.• mAG=100• mCE=30• mEF=25• Find the measure
of all numbered angles.
Q
G
F
D
E
C
123
45
6
A
R
S
P
Q
Inscribed Quadrilaterals• If a quadrilateral is inscribed in a circle,
then the opposite angles are supplementary.
mPSR + mPQR = 180
