We have just proven the following theorem.
Theorem 12.2: In 
rectangles do not exist and
all triangles have angle sum less than 
.
This tells us that in hyperbolic geometry the defect of any triangle is a positive real number. We shall see that it is a very important quantity in hyperbolic geometry.
Corollary In all convex quadrilaterals have angle sum
less than 
.
Proof: Given any quadrilateral 
. Take the diagonal AC and
consider triangles 
and 
. By
Theorem 13.2 both of these triangles have angle sum less than
. The assumption that is convex implies that
and
. By adding all six angles we have
that the angle sum of is less that .