15. Complete the proof of Theorem 1. (ASsume B-A-C, then obtain a contradiction.)

Elementary Geometry For College Students, 7e
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ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
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ChapterP: Preliminary Concepts
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Please answer question 15 (2.4) Theorem is also attached. 

15. Complete the proof of Theorem 1. (ASsume B-A-C, then obtain a contradiction.)
Transcribed Image Text:15. Complete the proof of Theorem 1. (ASsume B-A-C, then obtain a contradiction.)
THEOREM 1: If A-B-C then C-B-A, and neither A-C-B nor B-A-C
PROOF
Given: A-B-C.
Prove: (a) C-B-A
(b) A-C-B and B-A-C are impossible.
CONCLUSIONS
JUSTIFICATIONS
(a)
(1) AB + BC = AC and A, B, C
Definition of betweenness
are distinct and collinear points.
(2) BC +AB = AC.
(3) СВ + BА - СА.
(4) :. С-В-А.
Algebra
Symmetry (Axiom D-3)
Definition of betweenness
CA.
(b)
Assumption for indirect argumen
(5) Assume A-C-B. Then
AC + СВ - АВ.
(6) (AC+CB) + BC = AC
(7) 2BC = 0, or BC = 0 and
B = C
Substitution: Steps (1) and (5)
Algebra (B # C)
%3D
(The case B-A-C is similar, and is left as Problem 15.)
SI
Transcribed Image Text:THEOREM 1: If A-B-C then C-B-A, and neither A-C-B nor B-A-C PROOF Given: A-B-C. Prove: (a) C-B-A (b) A-C-B and B-A-C are impossible. CONCLUSIONS JUSTIFICATIONS (a) (1) AB + BC = AC and A, B, C Definition of betweenness are distinct and collinear points. (2) BC +AB = AC. (3) СВ + BА - СА. (4) :. С-В-А. Algebra Symmetry (Axiom D-3) Definition of betweenness CA. (b) Assumption for indirect argumen (5) Assume A-C-B. Then AC + СВ - АВ. (6) (AC+CB) + BC = AC (7) 2BC = 0, or BC = 0 and B = C Substitution: Steps (1) and (5) Algebra (B # C) %3D (The case B-A-C is similar, and is left as Problem 15.) SI
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