 
Summary: Angle sums and more.
Among other things, we will prove the following Theorems.
Theorem One. If the angle sum of some triangle is then the angle sum of some right triangle is .
Theorem Two. If the angle sum of some triangle is then there is a rectangle.
Theorem Three. If there is a rectangle the the angle sum for every right triangle is .
Theorem Four. If the angle sum for every right triangle is then the angle sum for any triangle is .
Theorem Five. If the angle sum for every right triangle is then the parallel postulate holds.
Theorem Six. If the parallel postulate holds then the angle sum for every triangle is .
The proofs follow. Along the way we will prove some Theorems which are interesting in their own right.
Definition. Suppose T is a triangle. We let
(T)
be minus the sum of the angles of T. Don't ever forget that, by the Saccheri Legendre Theorem, (T) 0.
For some reason that escapes me the book calls (T) the defect of T. Does that mean there might be
something wrong with T if its "defect" (T) is positive?
Theorem. The "additivity of the defect" in the book. Suppose T is a triangle with vertices a, b, c
and d s(b, c). Let T and T be the triangles with vertices a, b, d and a, c, d, respectively. Then
(T) = (T ) + (T ).
Proof. Let A and A be the angles of T and T , respectively, corresponding to the vertex a. Let D
and D be the angles of T and T , respectively, corresponding to the vertex d. By the Crossbar Theorem,
A = A  + A . Thus
