An Axiom System for High School Geometry Based on Isometries

Affine Transformations algebraic alternate angles approach assumes axiomatic method axioms beginning CHAPTER circle common concept congruent consider considerable construct Continuity Corollary correspondence curriculum defined Definition dilation distance preserving transformations element equal Euclid's Euclidean geometry example existence fact familiar figures follows formations fundamental geometry course given Hence high school geometry Hilbert's importance included interesting intersection introduced intuition involve isometries L₁ L₂ logical maps material mathematics natural necessary parallel parallel lines pass path curve perpendicular plane possible postulate presentation problem proof properties proposal proposed course prove question real numbers reflection relationship require rigid motions rigor Role rotation segments serve side similarity transformation slide space standard statements straight line student suitable superposition teacher Theorem thesis tion topics trans translation triangle ABC turn University