Shaquille-
The lack of parallel lines has never seemed to be of any consequence - other than aesthetics - to me!
If you are dealing in a single plane - and there are multiple lines they all will intersect with each other.
In a three dimensional situation I believe you could have an infinite number of non-parallel lines which could all intersect at one point - or could all never intersect anywhere.
(Is that vague enough to sound intelligent?)
RICHARD
You're living in a Non-Euclidean world? Write a more comprehensible question.
Can't eplain it at all, Lucy.