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Solve for the measure of angle JKL in the diagram below? Show your work and explain the steps you used to solve. 80

Solve for the measure of angle JKL in the diagram below? Show your work and explain the steps you used to solve. 80

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Solve for the measure of angle JKL in the diagram below. Show your work and explain the steps you used to solve. In the diagram, we have: - A circle with a tangent line at point L. - Line segment KJ forming an angle at point J (76° with the tangent line). - LM is a tangent to the circle. - Line segment KM intersecting the circle at L, forming angle KLM (50°). **Steps to Solve:** 1. **Identify the known angles:** - LJM = 76° (angle between line segment LJ and the tangent). - KLM = 50° (inscribed angle opposite to the arc). 2. **Find the angle JLM:** - As we know, the angle between the tangent and the chord through the point of contact is equal to the angle subtended by the chord in the alternate segment of the circle. - Therefore, angle JLM = angle KLM = 50°. 3. **Calculate angle JKL:** - By considering ∆ JKL, where JLM and LJM are known: - The sum of angles in a triangle is 180°. - Hence, angle JKL = 180° - (angle JLM + angle LJM). - So, angle JKL = 180° - (50° + 76°) = 54°. Therefore, the measure of angle JKL is 54°.
Transcribed Image Text:Solve for the measure of angle JKL in the diagram below. Show your work and explain the steps you used to solve. In the diagram, we have: - A circle with a tangent line at point L. - Line segment KJ forming an angle at point J (76° with the tangent line). - LM is a tangent to the circle. - Line segment KM intersecting the circle at L, forming angle KLM (50°). **Steps to Solve:** 1. **Identify the known angles:** - LJM = 76° (angle between line segment LJ and the tangent). - KLM = 50° (inscribed angle opposite to the arc). 2. **Find the angle JLM:** - As we know, the angle between the tangent and the chord through the point of contact is equal to the angle subtended by the chord in the alternate segment of the circle. - Therefore, angle JLM = angle KLM = 50°. 3. **Calculate angle JKL:** - By considering ∆ JKL, where JLM and LJM are known: - The sum of angles in a triangle is 180°. - Hence, angle JKL = 180° - (angle JLM + angle LJM). - So, angle JKL = 180° - (50° + 76°) = 54°. Therefore, the measure of angle JKL is 54°.
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