14 1 inscribed angles and intercepted arcs

Using Inscribed Angles & Polygons; Justifying Measurements & Relationships in Circles

14-1Inscribed Angles


Inscribed Angle: An angle whose

vertex is on the circle.

INSCRIBEDANGLE

INTER

CEP

TED

ARC


Name the intercepted arc for the angle.

C

L

O

T1.

CL


Name the intercepted arc for the angle.

Q

R

K

V2.

QVRS


2

ArcdIntercepteAngleInscribed

160º

80º

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

How do we solve for y?

The measure of the inscribed angle is HALF the measure of the inscribed arc!!

120

x

What is the value of x?

y

How do we solve for y?

The measure of the inscribed angle is HALF the measure of the inscribed arc!!

Since we know that the measure of x AND the

measure of y must both equal half of 120, then we

know that x=y

120/2 = 60

X= 60 Y= 60


Examples

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

M

Q

K

S

J

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

40

112


72º

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

Therefore we can say that the blue angle and the red angle

have the same angle measurement


Example 5

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

3

Q

D

JT

U

4Find m 4

Find arc QD

Find arc QTD


Example 5


3

Q

D

JT

U

4

Since we know that angle 3 and 4 intersect the same arc, we know that they must be

congruent, so we can set them equal to one another to find x.

TRY IT!


Example 5


3

Q

D

JT

U

4


180º

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

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

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14 1 inscribed angles and intercepted arcs

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