Drawing Triangles

Date: 06/18/97 at 10:52:20
From: Joe
Subject: Geometry (Triangles)

Is it possible to draw a triangle with more than 180 degrees?  

Thanks!  :-)

Date: 06/19/97 at 15:54:16
From: Doctor Daniel
Subject: Re: Geometry (Triangles)

Hi Joe,

If we suppose that we're drawing triangles on a "flat sheet of paper, 
it's impossible to draw a triangle with more than 180 degrees. I've 
drawn a picture for you that may help:
  
           

Suppose that we have a triangle with corners A, B, and C.  You've 
probably learned that we can draw a line through A parallel to BC; 
that's called the "parallel postulate" in some geometry books. Now, 
suppose we consider angle CAE. We know that it's the same measure as 
angle BCA, which is in the triangle. (That's by another rule for 
parallel lines.) Also, we know that angle DAB is the same measure as 
angle ABC, for the same reason.  

So, we see that mDAB + mBAC + mCAE = mABC + mBAC + mACB (where m means 
measure). But the first half of that equation equals 180 degrees, 
since three angles on the same side of a line fill that half of the 
line (That is, they add up to a "straight angle" of 180 degrees.)

Thus, the angles in the triangle also add to 180 degrees: on a flat 
sheet of paper, a triangle has exactly 180 degrees.

However, in a different geometry, a triangle might have more degrees. 
Consider a "triangle" on a globe that includes the north pole and 
two points on the equator. Here the two angles on the equator will be 
90 degrees, plus there'll be the angle at the north pole, which could 
be as much as 180 more degrees. In this case, the triangle might have 
more degrees than 180.

The difference is that on the sphere, the rule about parallel lines 
doesn't work, while it does on the flat plane.

This is just the very beginning of what's called "non-Euclidean 
geometry"; it gets much harder from here.  Your teacher may know 
something about it; if not, feel free to ask us more questions.  

Hope that helps!

-Doctor Daniel,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/