1. 1. 2x =
Solve for x.
ANSWER 58
2. x = ( 360 – 120)
1 2
ANSWER
120
– x = (( 2x + 4) + 28).
1 2
ANSWER 82
4. One-half the measure of an angle and its supplement is equal to the measure of the angle. Find the measure of the angle.
ANSWER
120º

2. Apply angle relationships in circles.
Target
Apply angle relationships in circles.
You will…
Find the measures of angles inside, on, or outside a circle.

3. Vertex Inside the Circle
Vocabulary
Angles and their Arcs
angle = sum of arcs 12
Vertex Inside the Circle
Central Angle –
Theorem –
50º
Extending the radii allow you to think of angles whose vertex is inside the circle the same way.

4. Vocabulary
Angles and their Arcs
angle = arc 12
Vertex On the Circle
Theorem 10.7 – Inscribed Angle
Theorem –

5. Vocabulary
Angles and their Arcs
angle = difference of arcs 12
Vertex Outside the Circle
Theorem –

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

7. GUIDED PRACTICE
for Example 1
Find the indicated measure.
SOLUTIONS = 12
(210o)
m XY = 2
(80o)
m 1
m RST = 2
(98o)
= 196o
= 160o
= 105o

8. 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.

9. 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.

10. 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.)

11. 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.

12. GUIDED PRACTICE
for Examples 2, 3, and 4
Find the value of the variable. 5. 6.
y = 61o
ANSWERS
= 104o a xo
253.7o