Jeremy Gibbs

Modern Geometry

Chapter 3, #1, #2

  1. Given A*B*C and A*C*D.
  1. Prove that A, B, C, and D are four distinct points.

According to the given, A*B*C. By Betweenness Axiom 1, A, B, and C are three distinct points. By the same axiom we can deduce that A, C, and D are all three distinct points.
  1. Prove that A, B, C, and D are collinear.

By BA1, if A*B*C, then A,B, and C are three distinct points lying on the same line. The same can be said for the other case, A*C*D. By BA1, A, C, and D are also distinct collinear points. Thus, A, B, C, and D are collinear.
  1. Prove the corollary to Axiom B-4.

COROLLARY. If A and B are on opposite sides of l and B and C are on the same side of l, then A and C are on opposite sides of l.

 
  1. a. Finish the proof of Proposition 3.1 by showing that ray AB U ray BA = line AB.

By definition of a ray and a line, ray AB ( line AB. Also, by the same definition, ray BA ( line AB. Let C be a point that is the union of ray AB and ray BA; we wish to show that C belongs to line AB. If C equals A, or if C equals B, then C is an endpoint of AB. Otherwise, A, B, and C are three distinct, collinear points (by definition of ray and Axiom 1), so exactly one of the relations A*C*B, A*B*C, or C*A*B holds (Axiom 3). If A*C*B, then C lies on segment AB, rays AB and BA, and line AB. If A*B*C, then C lies on ray AB, but not ray BA, and line AB. Finally, if C*A*B, then C lies on ray BA, not ray AB, but line AB. Thus, we can see that in any case, C lies on line AB.

  1. Finish the proof of Proposition 3.3 by showing that A*B*D.
  1. Prove the converse of Proposition 3.3 by applying Axiom B-1.

d. Prove the corollary to Proposition 3.3.