I was wondering whether there is a proof of SSS Congruence Theorem (and also whether there is one for SAS and ASA Congruence Theorem). In my textbook, they are treated as a postulate, or one that we just accept as truth without basis. Is the 3 theorems for similar triangles really just postulates, or are there any proof of them? I tried searching online but I couldn't find one.
The SAS criterion for congruence is generally taken as an axiom. From this, and using other postulates of Euclid, we can derive the ASA and SSS criterion. The proof proceeds generally by contariction.
For ASA criterion, we cut one of the sides so as to make it equal to corresponding part of the other triangle, and then derive contradiction.
Similarly, for SSS criterion, we arrive at contradiction by cutting one of the angles and making it equal ti the corresponding angle of the other triangle.