Parallel postulate

The Parallel Postulate: Given a line, and a point not on that line, there is one and only line that 1) passes through that point and 2) is parallel to the other line.

This is a foundation of Euclidean geometry, Euclid's fifth postulate (Actually I find it's a version of Playfair's Axiom, which is equivalent). It is, however, not necessarily so; coherent geometrical systems have been constructed under alternate assumptions, e.g.:

Hyperbolic geometry, or Lobachevskian geometry (or even Lobachevsky-Bolyai-Gauss geometry), says that there may be many lines that pass through our point and do not intersect our line.

Elliptic geometry, or Riemannian geometry, says that there are no such lines.

An example of the differences these systems may make in the "real world": Euclidean triangles have 180 degrees, Riemannian triangles have more than 180 degrees, and Lobachevskian triangles have less than 180 degrees, the difference depending on the size of the triangle.

THE PARALLEL POSTULATE

The Parallel Postualate states:
Through a point not in a line there is one and only one line parallel to the given line.

So, you probably want to know, what the heck does that mean? OK, get a piece of paper. Draw a line. Make a point (dot) somewhere above the line. Now draw a line through the dot. Say it's parallel to the other line. The Parallel Postulate says that such a line can exist, and only one of it can exist.

     ------0----------------- <-+
           |                    |
           |                parallel
           |                  |
------------------------- <---+

In Euclidean Geometry, a postulate is something you take for granted. It's something you can't prove, but something that needs be true for you to operate. A theorem is something you can prove using postulates and other theorems you've already proved. So, the fact that the Parallel Postulate is a postulate means that it can't be proved.

This is, however, incorrect. The Parallel Postulate can be proved, and is therefore a theorem, not a postulate. How can you prove it you ask? OK. Well, there's two ways you can prove something in geometry. Just prove it using things you already know, or assume that it's false, and then keep going along until you find that you're wrong in thinking it's false. The second method is called an indirect proof. It's the method I use to prove this theorem.

OK. Let's get started. First, we have to prove that that line you drew through that point can be parallel to the first line. So, erase the line you drew through that point. Now, you should just have a line with a point above it. Label the point P. Draw a line from the bottom line to P, making it at a right angle to the bottom line. Put a point on the right side of the vertical line. Label it K. Put a point on the left side of the vertical line. Label it O. Put a point even farther to the left. Label it Z.

   +->.     . <-P
   |        |      .<-K
   O  .<Z   |_
            | |
-------------------------

OK, there's this thing called the Angle Addition Postulate. Part of it says that all the rays (lines that go on forever in one direction only) that you can draw from point O on one side of the line ZK can be paired with the real numbers 1 through 180. This basically means your protactor will work.

Now, one of those rays we can draw from point O can be paired with the real number 90. That means it will have a 90° angle. Draw it. Put a point N on the end of that ray. You can put it anywhere, but it would look simpler if it was straight across from point P. OK, now, one of the rays that can be drawn from N can be paired with real number 90. Draw this ray, making sure it goes through point P.

Ray ON is perpendicular to line ZK. That means it intersects line ZK at a 90° angle. Ray NP is perpendicular to ray ON. Now, there is a theorem that states two lines (NP and ZP) that are both perpendicular to a third line (ON), are parallel to each other. This means ray NP is parallel to line ZK.

No, we're not done yet. Now we have to prove that there can be only one line going through point P that is parallel to ZK. It might seem obvious, but we have to prove it the nerdy mathematical way. But watch out, because this is where it gets confusing. To go on, you must forget everything you think you know about this picture except for what you can see, what I said in the last paragraph, and that all the lines are straight. You can't figure anything else out from just looking at the picture. OK? OK.

Now, put a point I between points O and N. Draw a ray from point I through point P. Say that the ray IP is perpendicular to the ray ON. That means it intersects it at a 90° angle. It might not look like it, but it's true, OK? OK. Now, due to the fact that two lines perpendicular to a third line are parallel to each other, ray NP is parallel to ray IP and line ZK. Ray IP is parallel to ray NP and line ZK. Line ZK is parallel to ray NP and ray IP.

Now, what was that little bit about parallel lines? They don't intersect, right? Exactly. That's part of the definition of parallel lines. However, ray NP and ray IP, which are parallel to each other, do intersect. But they can't. So only one of them can exist. Ray NP is unique. It is the only ray that can go through point P and be parallel to line ZK.

So, there you have it, a postulate that really isn't a postulate at all. Show your kids the paper you scribbled this on, and they'll think you are good at math after all.