Points, Lines, Angles, and Planes
How many ways are there to draw a segment on a piece of paper with both endpoints at corners of the page?
First, we'll make each of the corners a point, giving them names like A, B, C, and D. We know that for any two corners, there is exactly one segment connecting them, and those are the only segments we care about.
Writing the possible ways of choosing two corners, we get AB, AC, AD, BC, BD, and CD. These six segments are the four edges of the page and the two diagonals. Geometry is all about pictures, so here ya go.
What about lines? How many ways are there to draw lines using the four corners?
The answer is still six, but the lines have arrows at their ends because they extend to infinity and beyond. We can still write them as AB, AC, AD, BC, BD, and CD, though.
Some of the lines cross (BD and AC cross in the middle of the page) and some of the lines never cross (AD and BC, for instance). If lines on the same page never cross, we call them parallel, but we'll talk about them later. For now, we should give intersecting lines more attention.
When two lines cross, we say they intersect, which means they overlap at exactly one point. This makes sense if we draw it, since to get one line to cross another twice we end up having to curve the line or make a corner like this.
Lines are supposed to be straight, so that's a no go…or is it? After all, just because we can draw it doesn't make it logically true. We could draw a picture of a dragon eating a unicorn listening to Lindsay Lohan's "Speak," but that doesn't magically mean any living being actually bought that album.
This (the intersecting lines, not the unicorn-eating, Lohan-loving dragon) brings us to our very first proof in geometry: deducing logically that intersecting lines cross at exactly one point. We will use the "proof by contradiction" technique. Sounds snazzy, but it just means we're going to prove it by saying that the opposite can't possibly happen. Some call this an indirect proof because we're arriving at our conclusion indirectly.
Suppose that two different lines l and m cross at points A and B. Since any two points determine a line, we have l is the same as AB and m is the same as AB. But then that means l is the same as m, which contradicts our assumption that they were different lines.
Some prefer the "two column proof" format instead of the informal argument we used. We're going to be stuck introducing concepts for a while, so there aren't going to be many formal proofs right away. Don't be sad, though. We have a lot of more complicated proofs to look forward to later on.
All these points and lines remind us of connect-the-dots. Remember playing that as a kid? Well, if we imagine the most boring game of connect-the-dots imaginable, it would probably look something like this.
All we're really doing is extending the segment along the same straight line. This is what it means for points to be collinear: we can draw a single line through all of them at once. Of course, forming a line segment isn't exactly what we're signing up for when we play connect-the-dots. That's why the dots usually aren't collinear.
For example, there's no way of drawing a single line through points X, Y, and Z. (Don't try to draw a fat line, either because lines have no width or depth, only length.)
On the other hand, we can clearly see that the segment AC passes directly through B, so the points are collinear. Instead of saying "A, B, and C are collinear," we could say, "B is between A and C." They mean the same thing, but the second one tells us that B is in the middle.