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Chapter 3.3 CPCTC and Circles Megan O’Donnell 9 5/30/08 Objectives After studying this section you will be able to understand the following: The principle of CPCTC The basic properties of circles CPCTC CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent C P C T C CPCTC Explained In the diagram RIN DOG Therefore, we must draw the conclusion that R D This is because the angles are corresponding parts of congruent triangles, meaning they are exact replicas of each other. R N D I G O The Basics of Circles Point M is the center of the circle shown to the right. Circles are named by their center point. Thus, this circle is called Circle M. Circle M M Radii of Circles L E In a circle’s definition every point of the circle is equidistant from the center. A line reaching from the center to a point on the outside of a circle, such as LE is called a radius. Theorem 19 Theorem 19 states that all radii of a circle are Theorem 19 C A This means that CA LA L L Sample Problem Using CPCTC Statement Reason 1. 2. A D 1.Given 2 3 2.Given 3. 4. BE CE 3. If 5 6 then 4. Vertical angles are 5. ABC DCE 5. AAS (1,3,4) 6. AE DE 6. CPCTC Given: A D ; 2 3 Prove: AE DE Sample Problem With Circles Given: N Prove: QN RN Statement 1. Reason N Q 1.Given 2. QN RN 2.All radii of a are As simple as this!! M L NN N O P R Sample Problem With Both Ideas Statement 1. 2. Reason C 1. Given CD CE CB CA 3.BCA DEC 2.All radii of a are 3.Vertical angles Are 4.BCA DEC 4.SAS (2,3,2) 5. AB DE 5. CPCTC Given: C Prove: AB DE D B C A E Extra Problems Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. Given: WX WZ; VX VZ Prove: WXV WZV W Y Z X V ...More Statement Reason 1. 2. 3. 1. 2. 3. 4. 5. 6. 4. 5. 6. 7. 8. 7. 8. R RO MP M P O Given: Prove: C ; RO MP MR PR And More! Statement 1. Reason 1. 2. 2. 3. 3. A AD DC B 4. 4. 5. 5. 6. 6. C Given: B AD DC Prove: ABD CBD D ! ! ! ! And Even More!! ! Given: ! LM M = 3x+5 MN =6x-4 M N Find: x ! ! L Answers 1. WX WZ 2. VX VZ 3. WV WV 1.Given 2.Given 3.Reflexive 4. WXV WZV 4.CPCTC 1. C 1.Given 2. RO MP 2.Given 3. ROM ROP 3. Lines form right Right s 4.ROM ROP 4.Rt s are 5. 5.All radii of a MR PR are 6. RO RO 6.Reflexive ROM ROP 7.SAS (4,5,6) 7. 8. MR PR 8.CPCTC And more Answers Statement 1. 2. 3. C AD DC AB CB Reason 1.Given 2.Given 3.All radii of a Are 4. BD BD 4.Reflexive 5. ADB CDB 5.SSS (2,3,4) 6.ABD CBD 6.CPCTC 3x+5=6x-4 9=3x X=3 We can set these segments equal to each other because they are radii. We learned that all radii of a circle are congruent. Works Cited Fogiel, Matthew. Problem Solvers Geometry. Piscataway: Research and Education System, 2004. Milauskas, George, Richard Rhoad, and Robert Whipple. Geometry for Enjoyment and Challenge. Evanston: McDougal Littell,1991. The end! YAY GEOMETRY!