Now we need to consider a Saccheri quadrilateral which has base b, sides each with length a, and summit with length c. We showed that c>a, but we would like to know
Theorem 21.1: For a Saccheri quadrilateral
Figure: Saccheri Quadrilateral
Proof Compare Figure 22.1. Applying the Hyperbolic Law of
Cosines from Theorem 21.2, we have
From Theorem 21.1 we know that
Using these in Equation 22.1 we eliminate the variable d and have
Now, we need to apply the identity
and we have the formula.
Corollary: Given a Lambert quadrilateral ,
if c is the length of a
side adjacent to the acute angle, a is the length of the other side adjacent
to the acute angle, and b is the length of the opposite side,
then
Two segments are said to be complementary segments
if their lengths x
and 
are related by the equation
The geometric meaning of this equation is shown in the following figure,
Figure 22.2. These lengths then are complementary if the angles of
parallelism associated to the segments are complementary angles. This is then
an ``ideal Lambert quadrilateral'' with the fourth vertex an ideal point
.
Figure 22.2: Complementary Segments
If we apply the earlier formulas for the angle of parallelism to these
segments, we get
Theorem 21.2: [Engel's Theorem] There is a right triangle with sides and
angles as shown in Figure 22.3 if and only if there is a Lambert
quadrilateral with sides as shown is Figure 22.3. Note that PQ is a
complementary segment to the segment whose angle of parallelism is 
.
