Question

Prove True Fact 2: True Fact 1: If A-B-C and line L passes through B but not A, then A and C lie on opposite sides of L. TF1 is used to prove the following (in fact, the proof is not much different): True Fact 2: If point A lies on L and point B lies on one of the half-planes determined by L, then, except for A, the segment AB or ray AB lies completely in that half-plane.

Answer 1

If point A lies on L and point B lies on one of the half-planes determined by L, then, except for A, the segment AB or ray AB lies completely in that half-plane.

PROOF:

Let point A line on the line L and point B be in H1. Using the Ruler Postulate, we may find C such that A – C – B. Now C is in H1, on the line L, or in H2..

Suppose C is on L. This puts B on L since C and B are collinear by betweenness.

Thus we know that C is not on L.

Suppose C is n H1. Then, by Axiom D – 1, c. we have a point E on L with

A – C – E – B, which would put B on L by betweenness. Again, a contradiction.

So C is in H1.

A similar argument works for using the Ruler Postulate to find point F such

that A – B – F.

This puts both the segment AB and the ray AB in H1.

Q.E.D.