I'm hoping that someone can provide a method for deducing the commonly known SSS congruence postulate? The postulate states
If the three sides of one triangle are pair-wise congruent to the three sides of another triangle, then the two triangles must be congruent.
I am a middle school math teacher (teaching a HS Geometry course) and would like to be able to explain/justify the triangle congruence theorems that I expect students to apply with more clarity. I understand that the Law of Cosines could be used to justify the SSS triangle congruence theorem but I wonder if a proof can use more basic properties. My students are currently learning the congruence theorems and using constructions to justify their validity. I know two ways to demonstrate the validity of SSS congruence: constructions and the Law of Cosines. Is there another way?
I've researched this and have not seen another answer so far. The Law of Cosines is the accepted answer to Proof of ASA , SAS , RHS , SSS congruency theorem. I've also seen a few other questions hinting at this without a clear answer, see: Why is SSS criterion for congruence of triangles referred to as "SSS postulate" in textbooks?. Another question on math.SE asks what I'm asking (how can we derive the triangle congruence postulates from basic axioms?) Need for triangle congruency axioms and the accepted answer suggests only a quick visit to http://en.wikipedia.org/wiki/Taxicab_geometry but does not discuss whether or not the triangle congruence axioms follow from the basic axioms or are independent of them.