Give the reasons (axiom, definition, named theorem/prop. or general proposition) for each of the step in...

 

Transcribed Image Text:

Give the reasons (axiom, definition, named theorem/prop. or general proposition) for each of the step in the following proof: (2 points each) If points 0 AO' and radii OAA O'A', respectively determine the same circle y, then 0 = 0.A OA = 0'A'. a. OAO and OA^O'A' are center and radii of circle y b. Assume 0+ 0. c. Line 00 exists. d. 3 point C, C 0's.t. 00. C A OA=0'C e. O'A' = O'C f. C is on circle y so OA = OC g. OC O'C h. Either O. lies on CO or it's opposite ray i. Case 1: If O-on CO, then 0.= 0 (why) contradicting assumption. j. Case 2: If O. lies on ray opp. CO, then O.* C *O k. But this yields a contradiction. (why) 1. Thus 0 = 0. m. A, A. are on y so OA OA (why) and O. = 0 so OA = 0.A' 0' RAA (Contradiction) hypothesis same as part (e) RAA (Contradiction) conclusion same as part (f) Give the reasons (axiom, definition, named theorem/prop. or general proposition) for each of the step in the following proof: (2 points each) If points 0 AO' and radii OAA O'A', respectively determine the same circle y, then 0 = 0.A OA = 0'A'. a. OAO and OA^O'A' are center and radii of circle y b. Assume 0+ 0. c. Line 00 exists. d. 3 point C, C 0's.t. 00. C A OA=0'C e. O'A' = O'C f. C is on circle y so OA = OC g. OC O'C h. Either O. lies on CO or it's opposite ray i. Case 1: If O-on CO, then 0.= 0 (why) contradicting assumption. j. Case 2: If O. lies on ray opp. CO, then O.* C *O k. But this yields a contradiction. (why) 1. Thus 0 = 0. m. A, A. are on y so OA OA (why) and O. = 0 so OA = 0.A' 0' RAA (Contradiction) hypothesis same as part (e) RAA (Contradiction) conclusion same as part (f)