1. 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.

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

4. Find the indicated measure.
= 105o

5. Find the indicated measure.
= 196o

6. Find the indicated measure.
= 160o

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

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

9. Find each measure.
mSTU
= 83

10. Find each measure.

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

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

14. mABD
Find each angle measure.

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

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

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

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

20. Find the value of the variable.

22. Finding Arc Measures V U
Find
Step 1 Find
= 67 67

23. Continued V U
Step 2 Find

24. Step 2 Find
Find mLP
Step 1 Find

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

26. 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?

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

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

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

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

31. 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

32. 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.)

33. 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
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.