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B. A. O. C. AB is the intercepted arc of C. Inscribed Angles. The vertex of C is on circle O. The sides of C are chords of circle O. C is an inscribed angle. E. D. O. F. Polygons and Circles. A polygon is inscribed in a circle if all its vertices lie on the circle.

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Inscribed angles

B

A

O

C

AB is the intercepted arc of C .

Inscribed Angles

The vertex of C is on circle O.

The sides of C are chords of circle O.

C is an inscribed angle.


Polygons and circles

E

D

O

F

Polygons and Circles

A polygon is inscribed in a circle if all its vertices lie on the circle.

Circle O is circumscribed aboutDEF.


Example 1

A

B

BCD

Which angle intercepts DAB?

C

D

O

Example 1

Which arc does A intercept?

C

Is quadrilateral ABCD inscribed in the circle?

Yes

Which angles appear to intercept major arcs?

B and C

What kind of angles do B and C appear to be?

obtuse


Inscribed angle theorem

A

B

C

mB = ½ mAC

Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc:


Example 2

a°

P

mPRS = ½ mPS

mPQT = ½ mPT

T

30°

60°

S

Q

mPRS = ½ mPT + mTS

R

Example 2

Find the values of a and b:

60 = ½ a

a = 120

b = ½ (120 + 30)

b = ½ (150) = 75


Example 3

a°

P

mPRS = ½ mPS

mPQT = ½ mPT

T

25°

70°

S

Q

mPRS = ½ mPT + mTS

R

Example 3

Find the values of a and b:

70 = ½ a

a = 140

b = ½ (140 + 25)

b = ½ (165) = 82.5


Corollaries to the inscribed angle theorem

A

D

C

B

Corollaries to the Inscribed Angle Theorem

Corollary 1:

Two inscribed angles that intercept the same arc are congruent.

C  D


Corollaries to the inscribed angle theorem1

O

O

Corollaries to the Inscribed Angle Theorem

Corollary 2:

An angle inscribed in a semicircle is a right angle.


Corollaries to the inscribed angle theorem2

A

D

O

C

B

Corollaries to the Inscribed Angle Theorem

Corollary 3:

The opposite angles of a quadrilateral inscribed in a circle are supplementary.

A and C are supplementary.

B and D are supplementary.


Example 4

A

B

F

O

E

C

D

Example 4

Name a pair of congruent inscribed angles:

FAD and FBD

Name a right angle:

FBC

Name a pair of supplementary inscribed angles:

FED and FBD


Example 5

v°

96°

72°

Example 5

Find the values of of the variables:

v = 180 – 96 = 84

w = ½ (84) = 42

x = ½ (72) = 36

y = 180 – 72 = 108

z = ½ (84) = 42


Example 6

z°

107°

98°

Example 6

Find the values of of the variables:

w = ½ (107) = 53.5

x = ½ (180 – 98) = ½ (82) = 41

y = 180 - 107 = 73

z = ½ (98 + 73) = ½ (171) = 85.5


Angles formed by tangents and chords

B

D

C

B

mC = ½ mBDC

D

C

Angles Formed by Tangents and Chords

The measure of an angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc.


Example 7

A

58°

C

B

mCB = 2(32) = 64

D

Example 7

CD is tangent to circle O at C. AB is a diameter. Find the values of the variables.

x = 90 (the angle inscribed in a semicircle is a right angle)

y = 90 - 58 = 32

z = ½ (64) = 32


Example 8

J

Q

35°

z = ½ mJL = mQ = 35

L

K

Example 8

JK is tangent to circle O at J. QL is a diameter. Find the values of the variables.

x = 90

y = 90 - 35 = 55


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