5. (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 planes.
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Analytic Foundations of Geometry
Robert S. Wilson