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

32, 33] PARALLEL LINES ARE ASYMPTOTIC 57

Since the triangles KHM and KHN are congruent, and. HKP'=. HKQ, it easily follows that P'M is equal to QN.
But P' lies on the segment PM.
Therefore PM is greater than QN, and we have shown that
as we pass along the line a, in the direction of parallelism, the
distance from b continually diminishes.
We have now to prove the second part of the theorem.
Let a and b be two parallel lines as before, and P any point
on the line a.
M N M' b
FIG. 35.
Draw PM perpendicular to b, and let E be any assigned
length as small as we please.
If PM is not smaller than e, cut off MR =E.
Through R draw the ray a1 (RT) parallel to a and b in the
same sense.
Also draw through R the ray RS perpendicular to MR.
RS must cut the ray a; since z PRT is an obtuse angle.
Let it cut a at Q and draw QN perpendicular to b.
Now the lines RQS and the line b have a common perpendicular.
Therefore they are not-intersecting lines.
It follows that L NQR is greater than the angle of
parallelism for the distance QN.
At Q make LNQR'=L NQR.
Then L NQR' > z NQT', T' being any point upon PQ produced.