# 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 .

## a. Write a detailed version of the proof for Case 2.

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.

## b. Prove Case 3.

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.