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## PowerPoint Slideshow about 'Inscribed Angles' - galvin

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### Inscribed Angles

A

O

C

AB is the intercepted arc of C .

The vertex of C is on circle O.

The sides of C are chords of circle O.

C is an inscribed angle.

D

O

F

Polygons and CirclesA polygon is inscribed in a circle if all its vertices lie on the circle.

Circle O is circumscribed aboutDEF.

B

BCD

Which angle intercepts DAB?

C

D

O

Example 1Which 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

B

C

mB = ½ mAC

Inscribed Angle TheoremThe measure of an inscribed angle is half the measure of its intercepted arc:

a°

P

mPRS = ½ mPS

mPQT = ½ mPT

T

30°

60°

S

Q

b°

mPRS = ½ mPT + mTS

R

Example 2Find the values of a and b:

60 = ½ a

a = 120

b = ½ (120 + 30)

b = ½ (150) = 75

a°

P

mPRS = ½ mPS

mPQT = ½ mPT

T

25°

70°

S

Q

b°

mPRS = ½ mPT + mTS

R

Example 3Find the values of a and b:

70 = ½ a

a = 140

b = ½ (140 + 25)

b = ½ (165) = 82.5

D

C

B

Corollaries to the Inscribed Angle TheoremCorollary 1:

Two inscribed angles that intercept the same arc are congruent.

C D

O

Corollaries to the Inscribed Angle TheoremCorollary 2:

An angle inscribed in a semicircle is a right angle.

D

O

C

B

Corollaries to the Inscribed Angle TheoremCorollary 3:

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

A and C are supplementary.

B and D are supplementary.

B

F

O

E

C

D

Example 4Name a pair of congruent inscribed angles:

FAD and FBD

Name a right angle:

FBC

Name a pair of supplementary inscribed angles:

FED and FBD

v°

96°

w°

x°

72°

z°

y°

Example 5Find 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

z°

107°

98°

x°

w°

y°

Example 6Find 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

D

C

B

mC = ½ mBDC

D

C

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

y°

x°

58°

C

B

mCB = 2(32) = 64

z°

D

Example 7CD 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

Q

35°

x°

z°

y°

z = ½ mJL = mQ = 35

L

K

Example 8JK 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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