1. EXAMPLE 1
Find angle and arc measures
Line m is tangent to the circle. Find the measure of the red angle or arc.
SOLUTION = 12
(130o) a.
m 1
= 65o b.
m KJL = 2
(125o)
= 250o

2. GUIDED PRACTICE
for Example 1
Find the indicated measure.
SOLUTION = 12
(210o)
m 1
= 105o

3. GUIDED PRACTICE
for Example 1
Find the indicated measure.
SOLUTION
m RST = 2
(98o)
= 196o

4. GUIDED PRACTICE
for Example 1
Find the indicated measure.
SOLUTION
m XY = 2
(80o)
= 160o

5. Find an angle measure inside a circle
EXAMPLE 2
Find an angle measure inside a circle
Find the value of x.
SOLUTION
The chords JL and KM intersect inside the circle. xo = 12
(mJM + mLK)
Use Theorem xo = 12
(130o + 156o)
Substitute. xo
= 143
Simplify.

6. Find an angle measure outside a circle
EXAMPLE 3
Find an angle measure outside a circle
Find the value of x.
SOLUTION
The tangent CD and the secant CB intersect outside the circle.
m BCD
(mAD – mBD) = 12
Use Theorem = 12
(178o – 76o) xo
Substitute.
= 51 x
Simplify.

7. EXAMPLE 4
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.)

8. Solve a real-world problem
EXAMPLE 4
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.

9. GUIDED PRACTICE
for Examples 2, 3, and 4
4. Find the value of the variable.
y = 61o
ANSWER

10. GUIDED PRACTICE
for Examples 2, 3, and 4 5.
Find the value of the variable.
= 104o a
ANSWER

11. GUIDED PRACTICE
for Examples 2, 3, and 4
6. Find the value of the variable. xo
253.7o
ANSWER

12. Daily Homework Quiz 1.
Line t is tangent to the circle. Find the value of x and y.
ANSWER
115

13. Daily Homework Quiz
Line t is tangent to the circle. Find the value of x and y. 2.
ANSWER 52

14. Daily Homework Quiz
Line t is tangent to the circle. Find the value of x and y. 3.
ANSWER 25

15. Daily Homework Quiz
Line t is tangent to the circle. Find the value of x and y. 4.
ANSWER
x = 81; y = 40.5