This theorem gives us a setting for our later exploration into noneuclidean geometry.
Theorem 10.3:[Saccheri-Legendre Theorem]
The sum of the degree measures of
the three angles in any triangle is less than or equal to 
;
Proof: Let us assume not; i.e., assume that we have a triangle
in which 
. So there
is an 
so that
Figure 11.2: Saccheri-Legendre Theorem
Compare Figure 11.2. Let D be the midpoint of BC and let E be
the unique point on 
so that 
. Then by SAS
. This makes
Thus,
So, and 
have the same angle sum, even though
they need not be congruent. Note that
, hence
It is impossible for both of the
angles 
and 
to have angle measure greater than
, so at least one of the angles has angle measure greater
than or equal to .
Therefore, there is a triangle so that the angle sum is
but in which one angle has measure less than or equal to
. Repeat this construction to get another triangle with angle
sum but in which one angle has measure less than or equal to
. Now there is an 
so that
by the Archimedean property of the real
numbers. Thus, after a finite number of iterations of the above construction
we obtain a triangle with angle sum in which one angle has
measure less than or equal to 
Then the other
two angles must sum to a number greater than contradicting
Corollary 1 to Theorem 11.2.
Corollary 1: In the sum of the degree measures of two
angles is less than or equal to the degree measure of their remote exterior
angle.