Proof 11.23

Refer to the proof of Theorem 11.10 on page 278-279.

Suppose that

is a triangle in the hyperbolic plane. Let

be the midpoint of the segment

and let

be the midpoint of the segment

. Construct segments

and

that are perpendicular to the line

, with the points

lying on the line

Proof: Consider Case 2 where the point

is exterior to the segment

. Notice

and

. The total measure of the angles contained within Case 2 can be found by adding

and

. Notice,

[By construction]

. Notice that

is acute by construction. Therefore, the sum of the angles is less than

by definition.

Proof: Consider Case 3 where

equals point

(and thus equals point

). Notice

and

by construction. Then

by SSS and the angles contained within them are also congruent. With this, we can see

. By construction

and

are acute angles. Therefore, the sum of the angles is less than

We can conclude that each case is a Saccheri quadrilateral.