1. EXAMPLE 2 Find an angle measure inside a circle Find the value of x. SOLUTION The chords JL and KM intersect inside the circle. Use Theorem 10.12. xoxo = 1212 (mJM + mLK) xoxo = 1212 (130 o + 156 o ) Substitute. xoxo = 143 Simplify.

2. EXAMPLE 3 Find an angle measure outside a circle Find the value of x. SOLUTION Use Theorem 10.13. Substitute. Simplify. The tangent CD and the secant CB intersect outside the circle. = 1212 (178 o – 76 o ) xoxo = 51 x m BCD (mAD – mBD) = 1212

3. 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.)

4. EXAMPLE 4 Solve a real-world problem SOLUTION Use Theorem 10.13. Substitute. 149 o 1212 [(360 o – x o ) –x o ] Solve for x. xoxo 31 = 1212 m BCD (mDEB – mBD) 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°. ANSWER The measure of the arc from which the flash is visible is about 31 o.

5. GUIDED PRACTICE for Examples 2, 3, and 4 4. Find the value of the variable. SOLUTION The chords AC and CD intersect inside the circle. Use Theorem 10.12. Substitute. Simplify. = 1212 (y o + 95 o ) 78 o = y 61 78° (mAB + mCD) = 1212

6. GUIDED PRACTICE for Examples 2, 3, and 4 Find the value of the variable. 5.5. SOLUTION The tangent JF and the secant JG intersect outside the circle. Use Theorem 10.13. Substitute. Simplify. = 1212 (a o – 44 o ) 30 o = 104 a m FJG (mFG – mKH) = 1212

7. GUIDED PRACTICE for Examples 2, 3, and 4 6. Find the value of the variable. SOLUTION Use Theorem 10.13. Substitute. 73.7 o 1212 [(x o ) –(360 –x) o ] Solve for x. xoxo 253.7 = 1212 m TQR (mTUR – mTR) Because QT and QR are tangents, QR RS and QT TS Also,TS SR and CA CA. So, QTS QRS by the Hypotenuse-Leg Congruence Theorem, and TQS RQS. Solve right QTS to find that m RQS 73.7°.