1. Euclid stated his fifth and famous parallel postulate by saying that if two lines which were not parallel crossed a third that the angles on the side where the lines met would add up to less than 180o.
2. It is possible to prove this congruency criterion without using similar triangles. The two angles at the ends of the side will determine two lines
and where those two lines meet will determine the third point of the triangle.
This will even work in non-Euclidean spaces.
The next criterion (AAS) can also be proven without using similar triangles. One of the angles will be at one end of the side, but the other angle could be at any point on the line making the first angle with the given side.
Take one such line making that angle, and at the other end of the side, draw a line parallel to the line making the second angle.
Where that line meets the other side of the first angle is the third point of the triangle. However, this uses the parallel postulate and Theorem 6.4, and, in fact, this criterion is not valid in non-Euclidean spaces.