I've just started teaching congruent triangles to a class of 14/15 year olds in the UK. All that they are required to know for the purpose of national exams here is that two triangles are congruent if they have the same size and shape (obviously not a precise definition) and how to write out SSS / SAS / ASA / RHS proofs.
I have read in a few sources that congruence is often badly taught in schools, and that most students do not have a very strong grasp of what congruence is. For example, most students have no idea why SSS / SAS / ASA etc are sufficient for two triangles to be congruent. Maybe they can convince themselves by drawing pictures but that doesn't demonstrate they fully understand. This bothers me - what is the point of learning how to prove two triangles are congruent if one does not understand why the proof works?
Am I doing my students a disservice if I omit any discussion of why the proofs work? That's what most teachers here do. Is there a nice way to explain this without introducing lots of other precise definitions and proving the relevant theorems?