Simple Question, but I'm finding a lot of dispute on the "lesser" internet.
Basically, given a line, is it parallel with itself?
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It's sometimes hard for people learning mathematics, who naturally feel that mathematics is an objective discipline, to hear that many things are actually a matter of convention. This is one example of that, though. A quick scan of google books will show you that different authors use different definitions for "parallel", and that some of these definitions allow a line to be parallel to itself, while others don't.
There is a second issue in mathematical English that's relevant here. In advanced mathematics, when we say "two objects", we leave open the possibility that the two objects are actually equal. So for example, when I say "the sum of two even numbers is even" I am not requiring the numbers to be distinct. If I want the objects to be different I have to say "two distinct objects".
However, it appears to me that some of the geometry books I see on google books don't follow this convention. This isn't surprising to me, because
In any event, the variety of definitions and language conventions underscores the fact that you have to take the definitions of a book in the context of that book, and that you have to make sure that you understand the implicit language conventions (or lack thereof) used by the author.
Go back to other formulations of Euclid's Parallel postulate - for any point outside of a line there is at most one line that passes through this point and is parallel to this line. Clearly, here, there are as many lines that are parallel to a line as there are points that move away from that line (allowing us to ignore points on the same parallel lines). All the properties are defined, (such as angles adding up to 180\circ, etc.) but a line cannot be parallel to itself, as shown above and I hope here.
In short - a line cannot be parallel in relation to itself, as it IS itself.