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Inscribed Angles in Circles Bill Zahner Lori Jordan Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2016 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/about/ terms-of-use. Printed: January 24, 2016 AUTHORS Bill Zahner Lori Jordan www.ck12.org C HAPTER Chapter 1. Inscribed Angles in Circles 1 Inscribed Angles in Circles Here you’ll learn the properties of inscribed angles and how to apply them. What if your family went to Washington DC over the summer and saw the White House? The closest you can get to the White House are the walking trails on the far right. You got as close as you could (on the trail) to the fence to take a picture (you were not allowed to walk on the grass). Where else could you have taken your picture from to get the same frame of the White House? Where do you think the best place to stand would be? Your line of sight in the camera is marked in the picture as the grey lines. The white dotted arcs do not actually exist, but were added to help with this problem. After completing this Concept, you will be able to use inscribed angles to answer this question. Watch This MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/137536 CK-12 Foundation: Chapter9InscribedAnglesinCirclesA Learn more about inscribed angles by watching the video at this link. 1 www.ck12.org Guidance An inscribed angle is an angle with its vertex is the circle and its sides contain chords. The intercepted arc is the arc that is on the interior of the inscribed angle and whose endpoints are on the angle. The vertex of an inscribed angle can be anywhere on the circle as long as its sides intersect the circle to form an intercepted arc. Let’s investigate the relationship between the inscribed angle, the central angle and the arc they intercept. Investigation: Measuring an Inscribed Angle Tools Needed: pencil, paper, compass, ruler, protractor J 1. Draw three circles with three different inscribed angles. For A, make one side of the inscribed angle a diameter, J J for B, make B inside the angle and for C make C outside the angle. Try to make all the angles different sizes. 2. Using your ruler, draw in the corresponding central angle for each angle and label each set of endpoints. 3. Using your protractor measure the six angles and determine if there is a relationship between the central angle, the inscribed angle, and the intercepted arc. m6 LAM = m6 NBP = c = mLM c= mNP m6 QCR = c= mQR m6 LKM = m6 NOP = m6 QSR = Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc. 2 www.ck12.org Chapter 1. Inscribed Angles in Circles c If we had drawn in the central angle 6 ABC, we could also say that m6 ADC = In the picture, m6 ADC = 12 mAC. 1 6 2 m ABC because the measure of the central angle is equal to the measure of the intercepted arc. To prove the Inscribed Angle Theorem, you would need to split it up into three cases, like the three different angles drawn from the Investigation. Congruent Inscribed Angle Theorem: Inscribed angles that intercept the same arc are congruent. Inscribed Angle Semicircle Theorem: An angle that intercepts a semicircle is a right angle. In the Inscribed Angle Semicircle Theorem we could also say that the angle is inscribed in a semicircle. Anytime a right angle is inscribed in a circle, the endpoints of the angle are the endpoints of a diameter. Therefore, the converse of the Inscribed Angle Semicircle Theorem is also true. Example A c and m6 ADB. Find mDC c = 2 · 45◦ = 90◦ . m6 ADB = 1 · 76◦ = 38◦ . From the Inscribed Angle Theorem, mDC 2 Example B Find m6 ADB and m6 ACB. c Therefore, m6 ADB = m6 ACB = 1 · 124◦ = 62◦ The intercepted arc for both angles is AB. 2 3 www.ck12.org Example C Find m6 DAB in J C. Because C is the center, DB is a diameter. Therefore, 6 DAB inscribes semicircle, or 180◦ . m6 DAB = 12 · 180◦ = 90◦ . Watch this video for help with the Examples above. MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/137537 CK-12 Foundation: Chapter9InscribedAnglesinCirclesB Concept Problem Revisited You can take the picture from anywhere on the semicircular walking path. The best place to take the picture is subjective, but most would think the pale green frame, straight-on, would be the best view. Guided Practice c m6 MNP, m6 LNP, and mLN. c Find m6 PMN, mPN, 4 www.ck12.org Chapter 1. Inscribed Angles in Circles Answers: m6 PMN = m6 PLN = 68◦ by the Congruent Inscribed Angle Theorem. c = 2 · 68◦ = 136◦ from the Inscribed Angle Theorem. mPN m6 MNP = 90◦ by the Inscribed Angle Semicircle Theorem. m6 LNP = 12 · 92◦ = 46◦ from the Inscribed Angle Theorem. c we need to find m6 LPN. 6 LPN is the third angle in 4LPN, so 68◦ + 46◦ + m6 LPN = 180◦ . m6 LPN = To find mLN, ◦ c = 2 · 66◦ = 132◦ . 66 , which means that mLN Interactive Practice MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/113000 Explore More Fill in the blanks. 1. 2. 3. 4. An angle inscribed in a ________________ is 90◦ . Two inscribed angles that intercept the same arc are _______________. The sides of an inscribed angle are ___________________. J Draw inscribed angle 6 JKL in M. Then draw central angle 6 JML. How do the two angles relate? Find the value of x and/or y in J A. 5. 5 www.ck12.org 6. 7. 8. 9. Solve for x. 10. 6 www.ck12.org Chapter 1. Inscribed Angles in Circles 11. 12. 13. 14. Suppose that AB is a diameter of a circle centered at O, and C is any other point on the circle. Draw the line c Explain why D is the midpoint of through O that is parallel to AC, and let D be the point where it meets BC. c BC. 15. Fill in the blanks of the Inscribed Angle Theorem proof. Given: Inscribed 6 ABC and diameter BD c Prove: m6 ABC = 12 mAC TABLE 1.1: Statement 1. Inscribed 6 ABC and diameter BD m6 ABE = x◦ and m6 CBE = y◦ 2. x◦ + y◦ = m6 ABC 3. 4. 5. m6 EAB = x◦ and m6 ECB = y◦ 6. m6 AED = 2x◦ and m6 CED = 2y◦ Reason All radii are congruent Definition of an isosceles triangle 7 www.ck12.org TABLE 1.1: (continued) Statement c = 2x◦ and mDC c = 2y◦ 7. mAD 8. c = 2x◦ + 2y◦ 9. mAC 10. c = 2m6 ABC 11. mAC c 12. m6 ABC = 1 mAC Reason Arc Addition Postulate Distributive PoE 2 Answers for Explore More Problems To view the Explore More answers, open this PDF file and look for section 9.5. 8