Please answer question 18. Theorem is also attached. THEOREM 1: If A-B-C then C-B-A, and neither A-C-B nor B-A-CPROOFGiven: A-B-C.Prove: (a) C-B-A(b) A-C-B and B-A-C are impossible.CONCLUSIONSJUSTIFICATIONS(a)(1) AB + BC = AC and A, B, CDefinition of betweennessare distinct and collinear points.(2) BC +AB = AC.(3) СВ + BА - СА.(4) :. С-В-А.AlgebraSymmetry (Axiom D-3)Definition of betweennessCA.(b)Assumption for indirect argumen(5) Assume A-C-B. ThenAC + СВ - АВ.(6) (AC+CB) + BC = AC(7) 2BC = 0, or BC = 0 andB = CSubstitution: Steps (1) and (5)Algebra (B # C)%3D(The case B-A-C is similar, and is left as Problem 15.)SI 18. Prove that if A, B, and C are any three distinct, collinear points, then either A-B-C,A-C-B, or B-A-C.

Given A, B and C be three points that are distinct and collinear. Distinct means A≠B≠C and collinear…

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18. Prove that if A, B, and C are any three distinct, collinear points, then either A-B-C, A-C-B, or B-A-C.

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Question

Please answer question 18. Theorem is also attached.

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

18. Prove that if A, B, and C are any three distinct, collinear points, then either A-B-C,
A-C-B, or B-A-C.

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