Is it fair to say that plane geometry relies on the most fundamental level (ie apart from the postulates, eg Euclid's) on definitions of rotation - and thus on central symmetry? For example, opposite angles are formed by the intersection of two lines, have a common vertex and their sides are opposite halflines with that vertex as sole common point, thus they have central symmetry with it as center. But given it is very common (and useful, of course) to present progressions of theories in geometry which can arrive at other approachable theorems through rotation in a lot more round-about (unintended pun) ways (eg not present either sum of triangle's angles as 2 right angles by using a parallel line to the base which has the external to the base vertex in it=>and not using this leads to the external angle of a triangle not being quickly identified as the sum of the two that aren't its supplementary).
Asking because all sorts of progressions of theorems do highlight properties which (while no doubt equally important as any other) obviously wouldn't be that sought after in specific problems if a different progression was used - and yet my sense currently is that central rotation is incomparably the fastest route and more importantly appears to be on a fundamental level ubiquitous even if only tacitly a part of the other routes.
So I am primarily asking because it irks me a bit that even in progressions which explicitly don't wish to refer to that approach to get a faster result (fewer steps in the current proof), parts of it are
still there (like the opposite angles congruence).
To make more clear what I am talking about, I will give a very specific (and obv simple) example: you can prove that an external angle is larger than any of the two non-supplementary to it internal ones of the triangle, without explicitly using properties of rotation (180 degrees being the sum of internal angles isn't mentioned at all), but you would
still be tacitly using them (for opposite angle congruence).
Or is there a way to provide geometric proofs for this - and similar - without
at all using rotation?
It also irks me that in such approaches some elements are by and large taken for granted anyway (inevitably) due to visual presentation - for example that angle b1<angle b, which while evident, and also
provable if one wishes to do it, still makes the observer perhaps too involved in the actual system. I am sure, of course, that there are non-geometric approaches which do not make use of what is visually evident geometrically and thus other properties become apparent much like they do when parts of the rotation properties are not taken for granted.