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Comprehensive Guide to Solving Angles and Arc Measures in Geometry

This resource provides step-by-step solutions to various geometric problems involving angles and arc measures. From determining the value of x in equations such as 2x = 84 to exploring angle measures inside and outside circles, it utilizes theorems to simplify and find solutions. The guide also includes examples of real-world scenarios like calculating visible arcs during phenomena such as the Northern Lights. Practice problems and answers help reinforce understanding of key concepts necessary for mastering geometry.

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Comprehensive Guide to Solving Angles and Arc Measures in Geometry

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  1. Solve for x. 1. 2x = 84 + 32 1 2 58 ANSWER 2. x = ( 360 – 120) = 0 120 ANSWER

  2. 3.180 – x = (( 2x + 4) + 28). 1 2 82 ANSWER 45º ANSWER Solve for x. 4. One-half of the measure of an angle plus its supplement is equal to the measure of the angle. Find the measure of the angle.

  3. = a. m1 12 (130o) (125o) 2 = b. m KJL EXAMPLE 1 Find angle and arc measures Line mis tangent to the circle. Find the measure of the red angle or arc. SOLUTION = 65o = 250o

  4. = m1 12 (210o) for Example 1 GUIDED PRACTICE Find the indicated measure. SOLUTION = 105o

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

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

  7. The chords JLand KMintersect inside the circle. (mJM + mLK) xo = 12 12 xo (130o + 156o) = xo = 143 EXAMPLE 2 Find an angle measure inside a circle Find the value of x. SOLUTION Use Theorem 10.12. Substitute. Simplify.

  8. The tangent CDand the secant CBintersect outside the circle. (mAD – mBD) m BCD = 12 12 xo (178o – 76o) = x = 51 EXAMPLE 3 Find an angle measure outside a circle Find the value of x. SOLUTION Use Theorem 10.13. Substitute. Simplify.

  9. 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.) EXAMPLE 4 Solve a real-world problem SCIENCE

  10. 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 149o [(360o – xo) –xo] xo 31 The measure of the arc from which the flash is visible is about 31o. ANSWER EXAMPLE 4 Solve a real-world problem SOLUTION Use Theorem 10.13. Substitute. Solve for x.

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

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

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

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

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

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

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

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