On the congruence of triangles Sorry for the perhaps somewhat trivial question,  but are the criteria for the congruence of two triangles,  i.e. "side-angle-side", "side-side-side" and "angle-side-angle", taken as postulates or can they be proven? I've read Euclid's take on it,  but the use of "superposition" seems somewhat unsatisfactory and vague as a proof?! 
 A: But it's not a Postulate in Euclid. What did I miss in this alleged answer, if anything? I think you have to look at Euclid's Common Notions regarding the relation of Parts to the Whole. Also, in Euclid, you either have one triangle or two. Superposition, a modern notion, confuses this distinction. To understand Euclid better, one has to read and study Euclid's book, Euclid's Data, which involves how things are GIVEN in Geometry. "Data" is the Latin equivalent of our "Given." PS: For Euclid, the Sides, Angles, and Areas all have to be Equal(s), not the Same. Euclid does not have our modern notion of an Equivalence Relation; there is no Reflexivity for Euclid (a Triangle is not Equal to itself). So Congruence is an Anachronism imposed on Euclidean Geometry, if you mean by it Superposition (at the instant of Superimposition there's only One triangle!) - Figures are "given" in Position!
PS: Unfortunately, Euclid does not explicitly write that only a single point can occupy a single position, and that a single line (or the collection of all the points on it) occupies single potion(s) each. Had he done so, we could argue that superposition is impossible. For example, when two lines intersect how many points are there? One could argue two (one for each line). But notice that there is no proof in Euclid for that in either case. The point of intersection is assumed to belong to both lines. Logically, however there are two possibilities: (1) that there are two points (in the zero dimensional space where they interpenetrate one another as waves do). (2) Or that one point goes "under" the other as in the case of mathematical knots. So there is the sense that "superposition" is impossible in Euclidean Geometry. I am not sure if Rigidity (mathematical) captures this sense of impossibility. I believe that it is precisely these omissions in Euclid which made Dedekind's definition of real numbers possible, and in a broader sense resulted in the creation of mathematical subject of Topology by Henry Poincare and Felix Hausdorff
