Geometry Proofs with Lines

Date: 2/6/96 at 15:46:33
From: Anonymous
Subject: Geometric Proofs

Prove THEOREM 3.3: If two lines are cut by a transversal so that two 
interior angles on the same side of the transversal are supplementary,
then the lines are parallel.

Prove: If two lines are cut by a transversal so that alternate exterior 
angles are congruent, then the lines are parallel.

Date: 2/8/96 at 15:10:5
From: Doctor Ken
Subject: Re: Geometric Proofs

Hello!

For these questions, I'll assume that you're working in good old 
Euclidean Geometry, as in normal high school Plane Geometry and not 
Hyperbolic or Spherical Geometry.

For your first proof, try a proof by contradiction: assume that the two
lines _aren't_ parallel (that they meet at some point P), and then show
how that leads to a nonsense result.  Hint: let the two points where the 
transversal intersects the two lines be called A and B.  Then what is 
the sum of the angles in triangle ABP?

For the second one, you can actually moosh it into another form of the 
first proof: 

                          /
                         /x
------------------------/-------------------------------
                      y/
                    z /
---------------------/----------------------------------
                  x /
                   /

We know that the two angles labeled "x" are congruent.  Well, by 
vertical angles, we know that they have to be congruent to y, right?  
And the bottom x is supplementary to z, right?  See if you can show that 
that means that y and z are supplementary.  

Good luck!

-Doctor Ken,  The Math Forum