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

23, 24] SOME THEOREMS ON PARALLELS

45

In the region bounded by A'D' and A'H draw through A'
any ray A'P, and produce PA' through A' to Q.
Upon A'Q take any point R and join AR.
Then RA produced must intersect DC.
It follows that A'P must intersect D'C.
Therefore, as above, the ray A'H is a parallel through any
point A', on HA produced, to the line BC.
In both cases the parallels are in the same sense or direction as
the original ray (i.e. both right-handed or both left-handed).
We are thus entitled to speak of a line AB as a right-handed (or
left-handed) parallel to another line CD, without reference to
any particular point upon the line AB.
~24. Another property of parallels with which we are
familiar in Euclidean Geometry also holds for the Hyperbolic
Geometry.
If the line AB is parallel to the line CD, then the line CD is
parallel to the line AB.
From A draw AC perpendicular to CD; and from C draw CE
perpendicular to AB.
A
E
N
L ~ ~B
GH N K
C: D
FIG. 18.
In the region DCE draw any ray CF, and from A draw AG
perpendicular to CF.
It is easy to show that the point G must lie in the region
ECD.
Further, since L ACG is an acute angle and L AGC is a right
angle, AC> AG.