6.4Use Inscribed Angles and Polygons Theorem 6.9 Measure of an Inscribed Angle Theorem The measure of an inscribed angle is one half the measure of its

6.4 Use Inscribed Angles and Polygons

Theorem 6.9 Measure of an Inscribed Angle TheoremThe measure of an inscribed angle is one half the measure of its intercepted arc.

_____ 2

1ADB m AB D C

A

B


Example 1 Use inscribed anglesUse inscribed angles

Find the indicated measure in P.

P

Q

S

R

T

o60

o37

S a. m RQ b. mSolution

.______2

1_____

2

1S a. m RTm o60 o30

._______2__2QS b. mm R o73 o74Because RQS is a semicircle,

.________180_____180RQ o m QSm o74 o106

6.4 Use Inscribed Angles and PolygonsCheckpoint. Find the indicated measure. Checkpoint. Find the indicated measure.

GHJ 1. m

F

J

H

G

o100

______2

1GHJ m o100 o50

CD 2. m

D

B

Co40

______2CD m o40 o80


Theorem 6.10If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

_____ ADB m ACB

A

B

D

C


Theorem 6.11

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

o90ABC mAC

D

A

BC

if and only if ____ is a diameter of the circle.


Theorem 6.12A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

______GEFD mmmm o180

F

G

E

D

C

D, E, F, and G lie in C if and only if


Example 2 Use the circumscribed circleUse the circumscribed circleSecurity A security camera that rotates 90o is placed in a location where the only thing viewed when rotating is the wall. You want to change the camera’s location. Where else can it be placed so that the wall is viewed in the same way?SolutionFrom Theorem 6.11, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a ________ of the circle.

diameter

So, draw the circle that has the width of the wall as a _________. The wall fits perfectly with your camera’s 90 rotation from any point on the _________ in front of the wall.

diameter

semicircle

6.4 Use Inscribed Angles and PolygonsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

RTS 3. m

S

T

R

Qo31

By Theorem 6.10, if two inscribed angles of a circle intercept the same arc, then the angles are congruent. o31RTS m


4. A right triangle is inscribed in a circle. The radius of the circle is 5.6 centimeters. What is the length of the hypotenuse of the right triangle?

By Theorem 6.11, if a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

6.5r 6.52d cm 2.11


Example 3 Use Theorem 6.12Use Theorem 6.12

Find the value of each variable.

Solution

____RP mm o180

a. PQRS is inscribed in a circle, so its opposite angles are _____________.

a.

R

S

Q

Po100

o88

ox

oysupplementary

____SQ mm o180________ o yo100 o180

____y 80________ o xo88 o180

____x 92


Example 3 Use Theorem 6.12Use Theorem 6.12

Find the value of each variable.

Solution

____L_____ m o180

b. JKLM is inscribed in a circle, so its opposite angles are _____________.supplementary

____M_____ m o180____4____ o xo8x o180________ 180

____3____ o yo3y o180_______ 180

b.

L

M

K

J o8x

o3yo4x

o3y

Jm Km

x12____x 15

y6____y 30


. and of values theFind 5. ba

D

B

A o5a

Co96

o4b

180905 oo a

18a905 a

180964 oo b

21b844 b


Pg. 218, 6.4 #1-32

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6.4Use Inscribed Angles and Polygons Theorem 6.9 Measure of an Inscribed Angle Theorem The measure of an inscribed angle is one half the measure of its