warm-up

Warm-upFind the measure of each arc.

Homework Review

CCGPS Geometry10.21.14

UNIT QUESTION: What special properties are found with the parts of a circle?Standard: MMC9-12.G.C.1-5,G.GMD.1-3

Today’s Question:How do we use angle measures to find measures of arcs?Standard: MMC9-12.G.C.2

Inscribed Angles

Inscribed Angle: An angle whose

vertex is on the circle and

whose sides are chords of the circle

INSCRIBEDANGLE

INTER

CEP

TED

ARC

Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle.

C

L

O

T1.

YES; CL

Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle.

Q

R

K

V2. NO;

QVR

S

2

ArcdIntercepteAngleInscribed

160

80

To find the measure of an inscribed angle…

120

x

What do we call this type of angle?What is the value of x?

y

What do we call this type of angle?How do we solve for y?The measure of the inscribed angle is HALF the

measure of the inscribed arc!!

Examples

3. If m JK = 80, find m JMK.

M

Q

K

S

J

4. If m MKS = 56, find m MS.

40

112

72

If two inscribed angles intercept the same arc, then they are congruent.

Example 5

In J, m3 = 5x and m 4 = 2x + 9.Find the value of x.

3

Q

D

JT

U

4

x = 3

If all the vertices of a polygon touch the edge of the circle, the polygon is INSCRIBED and the circle is CIRCUMSCRIBED.

2 Column Proof

A circle can be circumscribed around a quadrilateral if and only if its

opposite angles are supplementary.

A B

CD

180 CmAm180 DmBm

z

y

110

85

110 + y =180y = 70

z + 85 = 180z = 95

Example 8 Find y and z.

180

diameter

If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle.

H

K

GN

4x – 14 = 90

Example 6

In K, GH is a diameter and mGNH = 4x – 14. Find the value of x.

x = 26

H

K

GN

6x – 5 + 3x – 4 = 90

Example 7

In K, m1 = 6x – 5 and m2 = 3x – 4. Find the value of x.

x = 11

1

2

Homework:

• Worksheet