In classical geometry why is a line considered to be parallel to itself? 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?

If two lines are both parallel to a third line, they should be parallel to each other, even if they are the same line.
