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Definition: Two lines are parallel if they are coplanar and everywhere equidistant.

Postulate 2: Through a point in a plane not on a line, one and only one line can be drawn parallel to that line.

Are parallel lines equal to each other, but there's some kind of disconnect I'm missing b/c they can't be one line?

So, I can't visualize this, and I feel he contradicts himself in Fact 3.

Fact 3: Parallelism is transitive: If $a \parallel b$ & $b \parallel c$, then $a \parallel c$.

(Fact 3 is why I questioned postulate 2 before reading fact 3 )

I probably need to read further, but if you can say something that might change my perspective, it'd be greatly appreciated. Thanks!

Najah

N. F. Taussig
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user593454
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  • Welcome to MathSE. Please read this MathJax tutorial, which explains how to typeset mathematics on this site. – N. F. Taussig Sep 15 '18 at 16:02
  • To answer your question, we would need to know how the author defines parallel lines. – N. F. Taussig Sep 15 '18 at 16:05
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    @N.F.Taussig He says they're: Two lines that are everywhere equidistant. Two parallel lines are always in the same plane – user593454 Sep 15 '18 at 16:08
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    The transitive property does imply that a line is considered parallel to itself. This does not contradict Postulate 2, which is about how many parallels exist that pass through a particular point not on the line. P2 doesn't state that that's the only way to get lines parallel to the original. (Now, one could augment Postulate 2 to state that there's one and only one parallel through any given point, noting that if the point is on the given line, then the parallel is the line. But postulates like to be as simple as possible. Separating the notions into Postulate 2 and Fact 3 is cleaner.) – Blue Sep 15 '18 at 16:28
  • @Blue Thank you, so much! Omg, so it's about how you can't have a pair of parallel lines on a point? I've heard this before, but the way I just said. Wow, thanks a million – user593454 Sep 15 '18 at 16:41
  • Let me fix that, in case anyone reads this later, If you have a line and a point. You can only draw 1 line through that point to be parallel to the first line – user593454 Sep 15 '18 at 16:51
  • @user593454: Correct. The one-and-only-one-parallel-through-a-point idea is a pivotal aspect of Euclidean geometry. To really blow your mind: it's possible to play the geometry game and deny this idea in one of two ways. If you declare that there are no parallels through a (separate) point, then you get something called "spherical geometry"; if you allow more than one parallel, then you get "hyperbolic geometry". And then all kinds of crazy stuff happens, like: the angles of a triangle do not add-up to $180^\circ$. Fun! :) Perhaps a bit advanced for you right now, but someday ... – Blue Sep 15 '18 at 19:22
  • @Blue Yeah! I'm excited for it! That's why I'm starting at square one. This really strengthened my morale. This community seems so cool! I can't wait to like... idk really appreciate a M.C. Escher. Is that how you say it? You call art by its creator. Idk, but thanks – user593454 Sep 15 '18 at 19:26

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