Angle sums and more. Among other things, we will prove the following Theorems. 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. De nition. Suppose T is a triangle. We let Æ(T ) be  minus the sum of the angles of T . Don't ever forget that, by the Saccheri Legendre Theorem, Æ(T )  0. For some reason that escapes me the book calls Æ(T ) the defect of T . Does that mean there might be something wrong with T if its \defect" Æ(T ) is positive? Theorem. The \additivity of the defect" in the book. Suppose T is a triangle with vertices a; b; c and d 2 s(b; c). Let T 0 and T 00 be the triangles with vertices a; b; d and a; c; d, respectively. Then Æ(T ) = Æ(T 0 ) + Æ(T 00 ): Proof. Let A 0 and A 00 be the angles of T 0 and T 00 , respectively, corresponding to the vertex a. Let D 0 and D 00 be the angles of T 0 and T 00 , respectively, corresponding to the vertex d. By the Crossbar Theorem, jAj = jA 0 j + jA 00 j. Thus Collections: Mathematics