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

46 NON-EUCLIDEAN GEOMETRY [CH. ImI.
From AC cut off AH =AG, and draw HK perpendicular to
AH on the same side as CD.
Make LHAL= GAB.
Then the ray AL must cut CD, and it follows that HK must
cut AL.
Let HK cut'AL at M.
From AB cut off AN =AM, and join GN.
Then the triangles HAM and GAN are congruent.
Thus z AGN =a right angle.
Therefore GN and GF coincide, and CF produced intersects
AB.
But CF was any ray in the region between CE and CD, and
CD itself does not cut AB.
Therefore CD is parallel to AB, in the same sense as AB is
parallel to CD.*
~ 25. A third important property of parallels must also be
proved:
If the line (1) is parallel to the line (2) and to the line (3),
the three lines being in the same plane, then the line (2) is also
parallel to (3).
Case I. Let the line (1) lie between (2) and (3). (Cf. Fig. 19.)
A
C D
BL
FIG. 19.
Let A and B be two points upon (2) and (3), and let AB cut
(1) in C..
Through A let any arbitrary line AD be drawn between
AB and (2).
* The proof in the text is adapted from that of Lobatschewsky in
New Principles of Geometry with a Complete Theory of Parallels, ~ 96
(Engel's translation, p. 169).