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By "parallelogram geometry" we mean the elementary, "commutative", geometrycorresponding to vector addition, and by "trapezoid geometry" a certain"non-commutative deformation" of the former. This text presents an elementaryapproach via exercises using dynamical software (such as geogebra), hopefullyaccessible to a wide mathematical audience, from undergraduate students andhigh school teachers to researchers, proceeding in three steps: (1)experimental geometry, (2) algebra (linear algebra and elementary grouptheory), and (3) axiomatic geometry.

Commutative and Non-commutative Parallelogram Geometry: an Experimental Approach