TITLE: In classical geometry why is a line considered to be parallel to itself?
QUESTION [33 upvotes]: A definition in classical geometry (for example, Birkhoff's formulation, but I suppose it could be all of them) is that a line is always considered to be parallel to itself. I understand this is probably for convenience, but in my mind since two distinct lines are parallel if they have no points in common and a line has infinitely many points in common with itself. Perhaps the idea is to ease the definition that two (non-parallel) lines intersect at one and only one point? 
Q: What's the purpose/what inconvenience would be caused if we didn't have that definition?

REPLY [2 votes]: The definitions of parallel vectors:

Two vectors $u$ and $v$ are parallel if their cross product is zero, i.e., $u\times  v=0$.
  Two vectors are parallel if they are scalar multiples of one another.

Similarly parallel lines can be characterized as: 

Two lines are parallel if they are scalar additions of one another. 

Obviously, in this sense the equal vectors as well as the same lines are parallel.