For this Next section we are going to Prove things Using Saccheri's and Lambert's Quadrilaterals.
Saccheri's Quadrilateral is a Quadrilateral with: a pair of parallel lines that are congruent, a base that is perpendicular to both parallel lines, and a summit connected to the top of the parallel lines. We do not know what the two summit angles are and with out using Euclid's Fifth Axiom, or one of the equivalent Theorems, there is no way to find out. Explore what you can find out without using any of the Theorems from Chapter 2.

A Lambert Quadrilateral is on such that at least three Angles are Right Angles.

Theorem 3.1:

The diagonals of a Saccheri quadrilateral are congruent.

Theorem 3.2:

The summit angles of a Saccheri quadrilateral are congruent.

Theorem 3.3

The summit angles of a Saccheri quadrilateral are not obtuse and thus are both acute or both right.

Theorem 3.4:

The line joining the midpoints of both the summit and the base of a Saccheri quadrilateral is perpendicular to both.

Theorem 3.5:

The summit and base of a Saccheri quadrilateral are parallel.

Theorem 3.6

In any Sacherri quadrilateral the length of the summit is greater than or equal to the length of the base.

Theorem 3.7

The fourth angle of a Lambert quadrilateral is not obtuse and thus is acute or right.

Theorem 3.8

The measure of the line joining the midpoints of the base and the summit of a Saccheri quadrilateral is less than or equal to the measure of its sides.

The diagonals of a Saccheri quadrilateral are congruent.

Theorem 3.2:

The summit angles of a Saccheri quadrilateral are congruent.

Theorem 3.3

The summit angles of a Saccheri quadrilateral are not obtuse and thus are both acute or both right.

Theorem 3.4:

The line joining the midpoints of both the summit and the base of a Saccheri quadrilateral is perpendicular to both.

Theorem 3.5:

The summit and base of a Saccheri quadrilateral are parallel.

Theorem 3.6

In any Sacherri quadrilateral the length of the summit is greater than or equal to the length of the base.

Theorem 3.7

The fourth angle of a Lambert quadrilateral is not obtuse and thus is acute or right.

Theorem 3.8

The measure of the line joining the midpoints of the base and the summit of a Saccheri quadrilateral is less than or equal to the measure of its sides.