SuperpositionJohn T.BaldwinSuperpositionJohn T. BaldwinOctober 22, 2007SuperpositionJohn T.BaldwinCongruence and Similarity1 Another take on Congruence2 Similarity -the role of arithmetic and geometrySuperpositionJohn T.BaldwinCongruence of triangles-HilbertBasic notions: point, line, incidence, between, congruence(angle,segment)defined terms: segment angleAxioms included SASSuperpositionJohn T.BaldwinCongruence of triangles-tranformation approachA transformation of a geometry is a permutation of the pointsand a permutation of the lines that preserves incidence.If it also preserves congruence of segments and angles then it isa rigid motionNow a key axiom is:Superpos ition Axiom:If angle BAC = DEF there is an isometry taking A to E andsuch that B0A0(i.e. BE ) lies on DE and C0A0lies on FE .yields SASSuperpositionJohn T.BaldwinExercisesExercise: Prove SAS from this axiom and Hilbert order andcongruence up to but not including SAS.Exercise: Prove that if AB∼=CD there is rigid motion mappingA to C and B to D.SuperpositionJohn T.BaldwinCongruence-WeinzweigBasic notions: point, line, incidence, between, motiondefined terms: segment angle, congruenceDefinition. Two figures are congruent if they are mapped toeach other by a motion.Axioms guarantee that the group of motions are ‘the rightgroup’:SAS is a