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.
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.
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.
Get Answers For Free
Most questions answered within 1 hours.