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Apply Other Angle Relationships in CirclesPowerPoint Presentation

Apply Other Angle Relationships in Circles

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### Apply Other Angle Relationships in Circles

Sunday, September 14, 2014

Essential Question:

How do we find the measures of angles inside or outside a circle?

Lesson 6.5

M2 Unit 3: Day 5

3.180 – x = (( 2x + 4) + 28).

1 2

82

ANSWER

45º

ANSWER

Warm Ups

Solve for x.

4. One-half of the measure of an angle plus its supplement is equal to the measure of the angle. Find the measure of the angle.

Theorem 6.13

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

a.

m1

12

(130o)

(125o)

2

=

b.

m KJL

Find angle and arc measures

EXAMPLE

Line mis tangent to the circle. Find the measure of the red angle or arc.

SOLUTION

= 65o

= 250o

Theorem 6.14 Angle Inside the Circle

If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arc intercepted by the angle and its vertical angle.

The chords JLand KMintersect inside the circle.

(mJM + mLK)

xo

=

12

12

xo

(130o + 156o)

=

xo

= 143

EXAMPLE 2

EXAMPLE

Find the value of x.

SOLUTION

Use Theorem 6.14.

Substitute.

Simplify.

The chords ACand CDintersect inside the circle.

=

12

12

78o

(yo + 95o)

=

61

= y

Guided Practice

4. Find the value of the variable.

SOLUTION

(mAB + mCD)

78°

Use Theorem 6.14

Substitute.

Simplify.

Theorem 6.15Angle Outside the Circle

Theorem 6.15 Secant and Tangent

Theorem 6.15Two Tangents

Theorem 6.15 Two Secants

The tangent CDand the secant CBintersect outside the circle.

(mAD – mBD)

m BCD

=

12

12

xo

(178o – 76o)

=

x

= 51

EXAMPLE

Find an angle measure outside a circle

Find the value of x.

SOLUTION

Use Theorem 10.13.

Substitute.

Simplify.

The tangent JFand the secant JGintersect outside the circle.

(mFG – mKH)

m FJG

=

12

12

30o

(ao – 44o)

=

a

= 104

GUIDED PRACTICE

Guided Practice

Find the value of the variable.

5.

Use Theorem 10.13.

Substitute.

Simplify.

Because QTand QRare tangents,

Also,TS SR and CA CA . So, QTS QRS by the Hypotenuse-Leg Congruence Theorem, and

(mTUR – mTR)

m TQR

QR RS and QT TS

12

12

TQS RQS.Solve rightQTS to find thatm RQS

73.7°.

=

73.7o

[(xo) –(360 –x)o]

xo

253.7

GUIDED PRACTICE

Guided Practice

6. Find the value of the variable.

SOLUTION

Use Theorem 10.13.

Substitute.

Solve for x.

The Northern Lights are bright flashes of colored light between 50 and 200 miles above Earth. Suppose a flash occurs 150 miles above Earth. What is the measure of arc BD, the portion of Earth from which the flash is visible? (Earth’s radius is approximately 4000 miles.)

EXAMPLE 4

Solve a real-world problem

SCIENCE

Because between CBand CDare tangents,

Also,BC DC and CA CA . So, ABC ADC by the Hypotenuse-Leg Congruence Theorem, and

BCA DCA.Solve rightCBA to find thatm BCA

74.5°.

(mDEB – mBD)

m BCD

=

CB AB and CD AD

12

12

149o

[(360o – xo) –xo]

xo

31

The measure of the arc from which the flash is visible is about 31o.

ANSWER

EXAMPLE 4

Solve a real-world problem

SOLUTION

Use Theorem 10.13.

Substitute.

Solve for x.

Homework between

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