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 10.12. xo = 12 (130o + 156o) Substitute. xo = 143 Simplify.

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 10.13. = 12 (178o – 76o) xo Substitute. = 51 x Simplify.

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.)

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 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.

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

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

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