Preliminaries

A polygonal region is a plane figure which can be expressed as the union of a finite number of triangular regions, in such a way that if two of the triangular regions intersect, their intersection is an edge or a vertex of each of them.

Let R be a polygonal region. A triangulation  of R is a finite collection,

of triangular regions , such that

1. the 's intersect only at edges and vertices, and
2. their union is R.

Let and be polygonal regions. Suppose that they have triangulations

such that for each i we have . Then we say that and are equivalent by finite decomposition , and we write .

Let for any triangular region T.

Theorem 17.1: If and are triangulations of the same polygonal region R, then .

Theorem 1721: If two Saccheri quadrilaterals have the congruent summits and equal defects, then their summit angles are congruent, in which case, the two Saccheri quadrilaterals are congruent. 

 
Figure 18.1: The Saccheri quadrilateral associated with

 

Given , with BC considered as the base.  Let D and E be the midpoints of AB and AC; let F, G, and H be the feet of the perpendiculars from B, A, and C, respectively, to . As you have proven in the homework, is a Saccheri quadrilateral. It is known as the quadrilateral associated with .  It depends on the choice of the base, but it should be clear which base we mean.

Theorem 17.3: Every triangular region is equivalent by finite decomposition to its associated quadrilateral region.

Theorem 17.4: Every triangular region has the same defect as its associated quadrilateral region.

Theorem 17.5: If and have the same defect and a pair of congruent sides, then the two triangular regions are equivalent by finite decomposition.

Theorem 17.6: [Bolyai's Theorem in the Hyperbolic Plane] If and are triangular regions, and , then .

Theorem 17.7: If , then there is a point P between A and C such that .