Geometry Proofs: Lines and Planes

Date: 11/09/98 at 22:21:14
From: Anonymous
Subject: Geometry proofs

Can you help me prove the following three theorems?

Theorem 1-1: If 2 lines intersect then they intersect in exactly one 
point.

Theorem 1-2: Through a line and a point not on the line there is 
exactly one plane.

Theorem 1-3: If 2 lines intersect, then exactly one plane contains the 
lines.

Please help me prove them. Thank you!

Date: 11/10/98 at 11:59:49
From: Doctor Peterson
Subject: Re: Geometry proofs

Hi Jaclyn, 

I don't know just what postulates and theorems you have to start with. 
Each book does things a little differently. But since you've been given 
these to prove yourself, you can guess that they can't be too hard, so 
everything you need is probably right there in the chapter. I'll give 
you the basic ideas, and you can fill in the details based on what you 
know.

1-1: You know that the lines intersect (in at least one point), so you 
need to prove that they can't intersect in two (or more) points. 
Suppose they did intersect at two points A and B. You probably have a 
postulate or theorem that there is only one line between any two 
points. Do you see how this tells you that what we've supposed is 
impossible?

1-2: You probably have a theorem or postulate that there is only one 
plane through three points that are not collinear. Given a line and a 
point, you can pick any two points on a line and you'll have three 
points to use. Now you have to prove that not only those three points, 
but the whole line is in the plane. You may have a theorem that already 
says that.

1-3: Again, if two lines intersect, you can pick three points to define 
a plane. You have to prove that both lines are in the plane. You can 
use what you proved in theorem 1-2 to show that both lines are in this 
plane.

If this isn't enough help, let me know what postulates and theorems 
immediately precede these. Those are probably what you will need to 
use.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/