inscribed angles 10.3 california state standards 4: prove theorems involving congruent and similar...

Inscribed Angles 10.3

California State Standards4: Prove theorems involving congruent and similar7: Prove/use theorems involving circles.16: Find probability given graph or table

21: Prove/solve relationships with circles.

definitionInscribed Angle

An angle whose vertex is on a circle and whose sides contain chords of the circle.

P

QR

definitionIntercepted Arc

An arc that lies in the interior of an inscribed angle and has endpoints on the angle.

P

QR

definitionsInscribed Polygon

A polygon with all of its vertices on a circle.

CircumscribeTo circle around something.

A

B C

D

EF

P

hexagon is inscribed in

ABCDEFP

circumscribeshexagon

PABCDEF

theoremInscribed Angle Measure

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

C

A

B

12m ACB mAB

D

example

B

C

140o

70o

Find the measure of the arc or angle.

A

mAC

C

theoremCongruent Inscribed Angles

Inscribed anglesare congruent.

B

A

E AAEB CB

that intercept the same arc

m ADE

m ACE

m ABE

example

CB

D

A

E

90o

45o

45o

45o

Find the measure of the arc or angle.

theoremInscribed Right Triangle

An inscribed right triangle has a diameter as its hypotenuse.

C

Q

P

R

is a right trianglePQR is a diameterPR

example

B

DA

C

Find the measure of the arc or angle.

180omADC

theoremInscribed Quadrilateral

A quadrilateral can be inscribed in a circle if and only if its opposite angles are

supplementary.

S

Q

P

R

180

180

m P m R

m Q m S

85 180y 95y

100x 80 180x

example

TS

R85o

U80o

xo yo

Find value of x.

example

B

A

C98o

196o

Find the measure of the arc or angle.

m ABC

D

m C

example

CB

D

44o44o

Find the measure of the arc or angle.

A

example

C

B

A

3xo

123 (180)x

30x

Find value of x.

_ _ _220 190 1800760 190 3420

x yx y

22 19 18040 10 180

x yx y

example

B

A

D

22xo

C40xo10yo

19yo

Find value of the measure of each angle.

3x 540 1620x

22(3) 66m D

40(3) 120m C

example

B

A

D

22xo

C40xo10yo

19yo

Find value of the measure of each angle.

180 114m B m D

180 60m A m C

120o

66o

60o

114o