In euclidean geometry we are used to having two triangles similar if their angles are congruent. It is obvious that we can construct two non-congruent, yet similar, triangles. In fact John Wallis attempted to prove the Parallel Postulate of Euclid by adding another postulate.
WALLIS' POSTULATE : Given any triangle 
and
given any segment DE. There exists a triangle 
having DE as
one of its sides that is similar to .
However, on page 124 of the text, it is proven that Wallis' Postulate is equivalent to Euclid's Parallel Postulate. Thus, we know that the negation of Wallis' Postulate must hold in hyperbolic geometry. That is, under certain circumstances similar triangles do not exist. We can prove a much stronger statement.
Theorem 12.3:[AAA Criterion]
In 
if 
, 
, and 
, then
. That is, if two triangles are similar, then
they are congruent.
Proof: Assume that 
. Then there
can be no corresponding sides which are congruent, else by ASA the
triangles would be congruent. Consider the triples 
and
. One of these triples contains at least two segments that are
larger than the corresponding segments of the other. Without loss of
generality, we can assume that
There exist points 
and 
such that 
and
. Then by SAS 
. Then
and 
. Thus, by the Alterate Interior Angles Theorem we can
show that 
does not intersect 
. Thus, it follows
that 
is convex. Using the fact that 
and 
are supplementary, as are 
and 
, the angle
sum of is 
, contradicting Corollary 1 to
Theorem 13.2.
As a consequence of Theorem 13.3 we shall see that in hyperbolic geometry a segment can be determined with the aid of an angle. For example, an angle of an equilateral triangle determines the length of a side uniquely. Thus in hyperbolic geometry there is an absolute unit of length as there is in elliptic geometry.