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## Warm Up

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**Apply Other Angle Relationships in Circles**Warm Up Lesson Presentation Lesson Quiz**3.180 – x = ((2x + 4) + 28)**1 2 1 2 82 58 ANSWER ANSWER 2. x = ( 360 – 120) = 0 120 ANSWER Warm-Up Solve for x. 1. 2x = 84 + 32**120**ANSWER Warm-Up 4. One-half of the measure of an angle plus the angle’s supplement is equal to the measure of the angle. Find the measure of the angle.**=**a. m1 12 (130o) (125o) 2 = b. m KJL Example 1 Line mis tangent to the circle. Find the measure of the red angle or arc. SOLUTION = 65o = 250o**ANSWER**105° Guided Practice Find the indicated measure. 1.m1**2.mRST**ANSWER 196° Guided Practice Find the indicated measure.**3.mXY**ANSWER 160° Guided Practice Find the indicated measure.**The chords JLand KMintersect inside the circle.**(mJM + mLK) x° = 12 12 x° (130° + 156°) = x = 143 Example 2 Find the value of x. SOLUTION Use Theorem 10.12. Substitute. Simplify.**The tangent CDand the secant CBintersect outside the circle.**(mAD – mBD) m BCD = 12 12 x° (178° – 76°) = x = 51 Example 3 Find the value of x. SOLUTION Use Theorem 10.13. Substitute. Simplify.**Example 4**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.)**Because CBand CDare tangents,**Also, BC DC and CA CA . So, ABC ADC by the Hypotenuse-Leg Congruence Theorem, and BCA DCA.Solve rightCBA to find thatm BCA 74.5°. (mDEB – mBD) m BCD = CB AB and CD AD 12 12 149° [(360° – x°) –x°] x 31 Example 4 SOLUTION Use Theorem 10.13. Substitute. Solve for x. The measure of the arc from which the flash is visible is about 31°.**ANSWER**y = 61 Guided Practice Find the value of the variable. 4.**ANSWER**a = 104 Guided Practice Find the value of the variable. 5.**ANSWER**x =about 253.7 Guided Practice Find the value of the variable. 6.**1.**ANSWER 115 Lesson Quiz Line t is tangent to the circle. Find the value of x and y.**ANSWER**52 Lesson Quiz Line t is tangent to the circle. Find the value of x and y. 2.**ANSWER**25 Lesson Quiz Line t is tangent to the circle. Find the value of x and y. 3.**ANSWER**x = 81, y = 40.5 Lesson Quiz Line t is tangent to the circle. Find the value of x and y. 4.