Sec. 12.4 Apply Other Angle Relationships in Circles p. 790 Objectives : Find the measures of angles formed by lines that intersect circles. Use angle measures to solve problems. Vocabulary: Review chord, secant, tangent Introduction: Why would a sailor climb to the top of a mast to watch for land? Today you will investigate how the height of the mast is related to the distance that can be seen.

Find angle and arc measures Line m is tangent to the circle. Find the measure of the red angle or arc. = 65o = 250o

Find the indicated measure. = 105o

Find the indicated measure. = 196o

Find the indicated measure. = 160o

Using Tangent-Secant and Tangent-Chord Angles Find each measure. mEFH = 65°

Using Tangent-Secant and Tangent-Chord Angles Find each measure.

Find each measure. mSTU = 83

Find each measure.

Find an angle measure inside a circle Find the value of x. xo = 143

Finding Angle Measures Inside a Circle Find each measure. mAEB = 126

mABD Find each angle measure.

Find each angle measure. mRNM mRNM = 22°

Finding Measures Using Tangents and Secants Find the value of x. = 40 = 63

Find the value of x. 50° = 83° – x x = 33°

Find an angle measure outside a circle Find the value of x. = 51 x

Find the value of the variable.

Finding Arc Measures V U Find Step 1 Find 180 -113 = 67 67

Continued V U Step 2 Find

Step 2 Find Find mLP Step 1 Find

Design Application In the company logo shown, = 108°, and = 12°. What is mFKH?

Two of the six muscles that control eye movement are attached to the eyeball and intersect behind the eye. If mAEB = 225, what is mACB?

Lesson Quiz: Part I Find each measure. 1. mFGJ 2. mHJK 41.5° 65°

Lesson Quiz: Part II 3. An observer watches people riding a Ferris wheel that has 12 equally spaced cars. Find x. 30°

Lesson Quiz: Part III 4. Find mCE. 12°

GUIDED PRACTICE 4. Find the value of the variable. y = 61o

QS = 5, QT = 4, so ëQST is a right ë. GUIDED PRACTICE 6. Find the value of the variable. QS = 5, QT = 4, so ëQST is a right ë. tan ÁTQS = ¾, so tan -1 (3/4) = 36.9à. 2 (36.9à) = ÁTQR = 73.7à Then, ÁTQR=1/2(x-(360-x)) 73.7 = 1/2(2x -360) 73.7 = x - 180 X = 253.7à

Solve a real-world problem SCIENCE 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.)

Use Theorem 10.13. Substitute. Solve for x. Solve a real-world problem SOLUTION Because CB and CD are tangents, CB AB and CD AD Also,BC DC and CA CA . So, ABC ADC by the Hypotenuse-Leg Congruence Theorem, and BCA DCA.Solve right CBA to find that m BCA 74.5°. = 12 m BCD (mDEB – mBD) Use Theorem 10.13. 149o 12 [(360o – xo) –xo] Substitute. xo 31 Solve for x. ANSWER The measure of the arc from which the flash is visible is about 31o.