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Understanding Circle Angle Measurements Through Practical Examples

This guide provides a comprehensive approach to solving problems related to angle measurements inside and outside a circle. Utilizing Theorems 10.12 and 10.13, we demonstrate how to find unknown angle values through substitution and simplification. Real-world applications like analyzing the visibility of flashes from the Northern Lights are included, showcasing the intersection of geometry and science. Guided practice problems reinforce the concepts and ensure understanding of the relationships between angles, tangents, and secants within circular geometry.

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Understanding Circle Angle Measurements Through Practical Examples

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  1. 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.

  2. 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.

  3. 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

  4. 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.

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

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

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

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