Let k and 
be lines in 
and let 
, 
be points on k. Let 
be the foot of in . We say that the
points 
are equidistant from if 
for all choices of i and j.
Theorem 13.8: If and 
are non-intersecting lines
in , then any set of points on equidistant from
has at most two elements.
Proof: Assume not; i.e., assume that 
are
equidistant from . Thus, the quadrilaterals 
, 
, and 
are all Saccheri quadrilaterals. Thus, 
, 
, and 
. Stringing these together, we have
Since 
and 
are supplementary, they are right
angles. Thus all three quadrilaterals are rectangles, a contradiction.
This theorem states that at most two points at a time on can be
equidistant from . It does not put a limit on the number of pairs of
such points, though. We may have pairs (A,B) and (C,D) so that 
and 
, but it is not possible that 
or any of
the other possibilities. It may also occur that the number of pairs of such
points will be zero. There is no guarantee that such a pair will exist. This
will help us distinguish types of parallel lines.
Theorem 13.9: Let and be non-intersecting lines
for which there is a pair of points 
which are equidistant from
. Then, and have a common perpendicular segment which is
the shortest segment between and .
Proof: is a Saccheri quadrilateral. Its altitude is
perpendicular to both and by Theorem 12.1. Also, the
quadrilateral formed by this altitude and any other segment from a point
and its foot 
is a Lambert quadrilateral. By
Theorem 12.6 the altitude is less than XX' and hence is the shortest
segment between and .
Theorem 13.10: In if lines and
have a common perpendicular segment MM', then they are non-intersecting and
MM' is unique. Moreover, if so that M is the midpoint of AB
then A and B are equidistant from .
Proof: We already know that the lines do not intersect from Corollary 1
to the Alternate Interior Angles Theorem. If PP' were another common
perpendicular then 
would be a rectangle, which cannot exist.
Thus, MM' is unique.
Let be so that M is the midpoint of AB. We have that
and 
, so by SAS 
. Thus, 
and 
. By Angle Subtraction 
. Then by
AAS 
and .
Lemma 13.2: Let MM' be the common perpendicular to and . Let
so that 
, then AA'<BB'.
If k and are non-intersecting lines which admit a common perpendicular
then they are said to be hyperparallel .
This is denoted by
. They are sometimes called divergently parallel
lines .
Recall that for 
there is a unique fan angle 
for P and . Let F be the foot of P in and let X and Y
be on opposite arms of the fan angle. The angle 
is
called the angle of parallelism at P with respect to and its
angle measure is denoted by 
. We know that its measure must be
less than 
.
Theorem 13.11: If is any line in and 
then there is a point so that 
.