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

31, 32]

NOT-INTERSECTING LINES

The converse is also true, namely, that
If two straight lines neither intersect nor are parallel, they
must have a common perpendicular.*
a I
B R N
b B' I R
FIG. 33.
Let a and b be the two'given lines, which neither intersect
nor are parallel.
From any two points A and P on the line a, draw AB and
PB' perpendicular to the line b.
If AB=PB', the existence of a common perpendicular
follows from ~ 28. Therefore we need only discuss the case
when AB is not equal to PB'.
Let PB' be the greater.
Cut off A' B' from PB' so that A'B' is equal to AB.
'At A' on the line A'B', and on the same side of the line as
A B, draw the ray a' making with A'B' the same angle as a, or
PA produced, makes with AB.
We shall now prove that a' must cut the line a.
Denote the ray PA by a1, and draw from B the ray h parallel
to al.
Since a, b are not-intersecting lines, the ray h must lie in the
region between BA and B'B produced.
Through B' draw the ray h', on the same side of B'A' as
h is of BA, and making the same angle with the ray B'B as
h does with B'B produced.
From ~ 26 (3), it follows that the parallel from B' to h and a,
lies in the region between h' and B' B.
* This proof is due to Hilbert; cf. loc. cit. p. 162.