10.3 Inscribed angles

Pg 613

Definitions• Inscribed angle- an whose vertex is on a

circle and whose sides contain chords of the circle.

• Intercepted arc- the arc that lies in the interior of the inscribed angle and has its endpts on the angle

A

B

C D

BD

(

BCD

Thm 10.8measure of an inscribed

• If an is inscribed in a circle, then its measure is ½ the measure of its intercepted arc.

mBAC= ½ m BC

A

B

C

(

ExampleExample

A

B

CD

E

m AED= ?(

180o

ExampleExample

A

CB

D

196o

mABD = ?98o

ExampleExample

A

B

C

D

70o

m AD = ?

(

140o

7xo

x = ?

70 = ½ 7x

Or

7x = 140

x=20

Thm 10.9• If two inscribed angles of a circle intercept

the same arc, then the s are

W

Y

C

X

Z

W Y

Inscribed Inscribed PolygonPolygon

• Polygon with ALL vertices on a circle.

A

B

C

D

ABCD is inscribed in the circle.

Circumscribed Circle – the circle around the inscribed polygon.

Thm 10.10Thm 10.10• If a rt. Δ is inscribed in a circle, then the

hypotenuse is a diameter of the circle.

• If one side of an inscribed Δ is a diameter of the circle, then the Δ is a rt. Δ & the opposite the diameter is the rt. .

A

BC

AC is a diameter.

If AC is a diameter, then ΔABC is a rt. Δ AND B is the rt. .

Ex: find x.Ex: find x.

PQ is a diameter,

R is a rt. .

3x = 900

x = 30

P

C

Q

R3xo

Thm 10.11Thm 10.11• A quadrilateral can be inscribed in a circle

iff its opposite s are supplementary.

A

B

C

D

mA + mC = 180o

mB +mD = 180o

So, can a rectangle be inscribed in a circle?

Yes, because its opposite s are supplementary.

Ex: x=? and Ex: x=? and y=?y=?

85 + x = 180

x = 95

80 + y = 180

y = 100xo

yo

80o

85o

Ex: x=? & y=?Ex: x=? & y=?

40x+10y=180

22x+19y=180

22x+19(18-4x)=180

22x+342-76x=180

342-54x=180

-54x=-162

x=3

10y 19y

22x 40x

4x+y=18

y=18-4x

y=18-4(3)

y=18-12

y=6

** Think back to Algebra!

Assignment