Elementary geometry has switched between different approaches many times. From synthetic to analytic, from axiomatic to structural etc.

Euclid took additivity of magnitude, e.g. area, as its starting point. Hilbert builds the theory of area from first principles. Euclid builds the number system from geometric data, whereas nowadays it is customary in Cartesian geometry to start from a number system and build the geometry. Classically, euclidean transformations are the ones that preserve lengths. Felix Klein showed how to build the euclidean invariants such as the length from the group itself. Area, length and angles are the primary concepts in the Elements. Nowadays the scalar product and orientation are taken as the basic definitions of an euclidean space in university courses.

It is quite interesting that what is an important theorem in a theory is close to being a definition in another one. I mentioned additivity of area. In the same fashion, the Pythagorean theorem is the last proposition of the first book of Euclid and its highest point. If one starts from a quadratic form, this is more or less a definition. Less known, Euclid actually proves that the plane has two dimension (prop. 7) from the additivity of angles. In modern approaches, a plan has two dimensions from its very definition and it is then deduced, sometimes painfully in cartesian coordinates, that the measure of angles is additive.

It is still a much heated debate which approach is the best with regards to teaching geometry to high school students.