Presentation 7.4 Inscribed Angles

8/8/2019 Presentation 7.4 Inscribed Angles

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Vocabulary

inscribed angle

intercepted arc


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An inscribed angle is an angle whose vertex

is on a circle and whose sides contain chords

of the circle.

intercepted

arc

The arc that lies in the interior of an inscribed

angle and has endpoints on the angle is called

the intercepted arc of the angle.

inscribed

angle


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Ifanangleisinscribedinacircle,thenits

measureis

C68 Inscribed AngleConjecture

m ADB = m AB1

2

C

A

BD

halfthe measureofits

interceptedarc.


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Find the measure of the QTS.

Finding Measuresof Arcsand Inscribed Angles

110

R

T Q

mQRS = mQTS

=1

2

= mQTS

220 = mQTS

S

C

110


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Find the measure ofQRS.


165mQRS = mQTS

=1

2

mQRS =( )

R

T Q

S

C

165mQRS = 82.5


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100

m NMP = 50

N

P

M 100C

Find the measure ofNMP.


m NMP = mNP

=1

2100

m NMP = ( )


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N

P

M

235

C

If mNMP = 235, then find the measure

ofNMP


125

m NMP = 62.5

m NMP = ( )

=1

2

mNP = 360 mNMP

mNP = 360 - 235

mNP = 125

m NMP = mNP


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W

X

Z

Find the measure ofZWX


m ZCX = m ZX

m ZWX =360 - 115

= m ZX115

m ZWX = 245

C

115

=

MAJOR ARC= 360 minorarc


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C69

A

D

CB

C

$ D

Iftwoinscribedanglesofa

circleinterceptthesamearc,

thentheanglesare

congruent.


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You decide that the

middle of the sixth row

has the best viewing

angle, point F.

THEATER DESIGN When you go to the movies, you want to be close to the

movie screen, but you dont want to have to move your eyes too much

to see the edges of the picture.

Usingthe Measureofan Inscribed Angle

moviescreenE G

F

IfEand Gare the ends of the screen and you are at F, mEFGis

called yourviewing angle.

If someone is sitting

there, where else can

you sit to have the

same viewing angle?


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SOLUTION

Draw the circle that is determined by the

endpoints of the screen and the sixth rowcenter seat.

Any other location on the circle will have the

same viewing angle.

Why?


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Ifalloftheverticesofapolygonlieonacircle,thepolygon

isinscribed inthecircleandthecircleiscircumscribed

aboutthepolygon.Thepolygonisaninscribedpolygonandthecircleisacircumscribedcircle. Thepolygonissaid

to be cyclic.


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A righttriangleisinscribedinacircleiffthehypotenuseisadiameterofthecircle.

C

A

B

C70

Forexample,

If ACisadiameterofthecircle,thenB isarightangleand

See GSP 10.5A

ifB isarightangle,then

ACisadiameterofthecircle.


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Ex: A, B, C, andDlieonacircle,

A

D

B

C

180mD +mB =

360mABC +mADC =


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A quadrilateralcanbeinscribedinacircleifand

onlyifitsoppositeanglesaresupplementary.

C

E

D

F

G

C71

Ex: D, E, F, and G lieonC,

iff m D + m F = 180 and

m E+ m G = 180.


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Is AC a diameter?ABCD is an inscribed quadrilateral inP.

C

B

D

53

P

mABC =

mCPD=

mBCD =

mADB =

mBCA =

mDBC =

180

26.5

307

63.5

106

127

?

If mBC = 53, then find ... A 53


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In the diagram, ABCD is inscribed in P.

Find the measure of each angle.

P

B

C

D

A

2y

3y

5x

3x


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ABCD is inscribed in a circle,

so opposite angles are

supplementary.

PB

C

D

A2y

3y

5x

3x

3x+

3y=

180 5x+

2y

=180

15x+ 6y= 540

-6x- 6y=

-360

*3

*-2

9x = 180

x =20

3(20) + 3y= 180

60 + 3y = 180

3y=

120y = 40


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x= 20 andy= 40

mA = 2y= 2(40)

= 80

mB = 3x

= 3(20)= 60

m

C = 5x= 5(20)

=100

m

D = 3y= 3(40)

= 120

PB

C

D

A2y

3y

5x

3x


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C72:Parallel lines interceptcongruent arcs on a circle.


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1. Givenacirclewith

centerpoint P.B

D

F

E

CA

H

P

G

2. Thecircleisdivided

into8congruent

arcs.

3. Thepointsoftheoctagonare A, B,

H,labeled

counter-clockwise.

4. Connectevery 3rd

pointwitha

diagonal, AD, DG,

GB, BEuntilyou

returntopoint A.


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mDAF = ( )90mDAF = 45

Find mDAF

360/8 = 45

45*2 = 90

F

mDAF = mDF

D

E

CB

A

H

P

G

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Presentation 7.4 Inscribed Angles