Why Do the Angles of a Triangle Add to 180 Degrees?

Date: 4/17/96 at 19:16:32
From: Anonymous
Subject: Geometry, angles in a triangle

Hi!  We were wondering why all the angles in a triangle 
add up to 180 degrees. Several of us are trying to prove 
Mr. Smith wrong and draw a triangle differently!

Thank you.

Date: 5/3/96 at 15:14:41
From: Doctor Steven
Subject: Re: Geometry, angles in a triangle

Have you ever tried drawing a triangle on a sphere?  I think you 
will then find the angles add up to more than 180 degrees.

For your problem, though, I think he wants triangles on a flat 
surface.

To see why the sum of the degrees of the angles on a triangle add 
to 180 look at a rectangle.  It has four angles each of 90 
degrees.  So the sum of its angles is 360 degrees.

Draw the diagonal of this rectangle and you will split the 
rectangle into two triangles.  Add up all the angles of the 
triangles and you should get 360 degrees since the rectangle had 
360 degrees (and we didn't expand the rectangle or anything). 

Your triangles are exactly alike, so they must have the same 
angles in them. Call the upper triangle T1 and the lower triangle 
T2. Then:

   sum of angles in T1 = sum of angles in T2.

   sum of angles in T1 + sum of angles in T2 = 360.

The only possible solution is that sum of angles of T1 = sum of 
angles of T2 = 180 degrees.

Note that you can create any right triangle just by changing the 
lengths of the sides of the rectangle.

To see this for other types of triangles, try using a rhombus or a 
trapezoid.

Hope this helps.

-Doctor Steven,  The Math Forum