The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

44

NON-EUCLIDEAN GEOMETRY

[CH. III.

In other words, if the ray AH is the right-handed (or lefthanded) parallel through A to the line BC, then it is the righthanded (or left-handed) parallel through any point upon the ray
AH, or HA produced, to the given line.
A A'
B D D C
FIG. 16.
Case I. Let A' be any point upon the ray AH other than A.
Through A' draw A'D' perpendicular to BC.
In the region bounded by A'D' and A'H draw any ray A'P,
and take Q any point upon A'P.
Join AQ.
Then AQ produced must cut DC.
It follows from Pasch's Axiom that A'Q must cut D'C.
But A'H does not cut D'C, and A'P is any ray in the region
D'A'H.
Therefore A'H is a parallel through A' to the line BC.
P
B D' D C
Flo. 17.
Case II. Let A' be any point upon the ray AH produced
backwards through A.
Draw A'D' perpendicular to BC.