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Wednesday, February 6, 2013Agenda• No TISK or MM• HW Questions (9-2 & 9-3)• Lesson 9-4: Inscribed Angles• Homework: Complete Ch9 HW Packet 1

§9-4 Inscribed Angles

Definition: Inscribed Angle

An angle with a vertex on the circle.

B

C

A

is an inscribed angleABC

§9-4 Inscribed Angles

Theorem:Measure of an Inscribed Angle

If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc.

B

C

A

12m ABC mAC

Practice: Find the measure of the indicated arc or angle.

A

B

85⁰D

Practice: Find the measure of the indicated arc or angle.

E

F

H160⁰

Practice: Find the measure of the indicated arc or angle.

J

L

N

70⁰

Example

In the circle shown, . Find and .

68mST

1

2

𝑚∠1=12𝑚ST

𝑚∠1=12(68 ° )

𝑚∠1=34 °

𝑚∠2=12𝑚ST

𝑚∠2=34 °

Theorem Theorem:

If two inscribed angles of a circle (or of congruent circles) intercept the same arc (or congruent arcs), then the angles are congruent.

ADC ABC

B

C

A

D

Inscribed Polygons

Definition: If all the vertices of a polygon

are located on a circle, then it is inscribed in the circle.

The circle is then circumscribed about the polygon.

B

C

A

is an inscribed quadrilateralABCD

D

Inscribed Polygons

Theorem:A right triangle is inscribed in a circle if and only if its hypotenuse is the diameter of the circle.

B

C

A

is a diameterCB is a rt.

Example

In , and Find , and X

4 5

6

1 2

3

is a rt. and (If a side of a triangle is the diameter of the circle, then it’s a right triangle. Def. rt. )

(same measure)

(corollary to Sum Th. & substitution)

(Substitution)

(Substitution)

( Sum Th. & Substitution)

Example

Quadrilateral QRST is inscribed in . If and , find and .

C

𝑚∠𝑆=12𝑚RT

28=12𝑚RT

56=𝑚RT

𝑚∠𝑄=12𝑚RST

𝑚=360−𝑚RST RT

𝑚=360−56RST

𝑚=304RST

𝑚∠𝑄=12(304)

𝑚∠𝑄=152

Example

Quadrilateral QRST is inscribed in . If and , find and .

C

𝑚∠𝑄=152

𝑚∠𝑆+𝑚∠𝑅+𝑚∠𝑇+𝑚∠𝑄=36028+110+𝑚∠𝑇 +152=360

𝑚∠𝑇+290=360𝑚∠𝑇=70

Theorem

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

B

C

A

D

are supp. are supp.

Practice. Find the value of each variable.

A

B C3x⁰

U

S

Tx⁰y⁰

R80⁰ 85⁰

Practice. Find the value of each variable.

A

B C3x⁰

U

S

Tx⁰y⁰

R80⁰ 85⁰