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Hilbert’s Axioms for Euclidean Geometry Axioms of Congruence. Jessica Erica Steven Information seen in this presentation is drawn from Roads to Geometry, Third Edition by Wallace and West. Background.
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Hilbert’s Axioms for Euclidean GeometryAxioms of Congruence Jessica Erica Steven Information seen in this presentation is drawn from Roads to Geometry, Third Edition by Wallace and West.
Background • David Hilbert was one of the most important figures in mathematics during the late 19th century and early 20thcentury. • In 1899 Hilbert published Grundlagen der Geometrie, a presentation of Euclidean geometry using an axiomatic system. • The axiomatic system was hoped to be consistent, independent, and complete. • Hilbert’s presentation of Euclidean geometry was free of shortcomings unlike Euclid’s.
Axiom1 • If & are 2 distinct points on line a • And if is a point on the same or another line • Then it is always possible to find a point on a given side of line through • Such that .
Axiom 2 • If • And if , • Then
Axiom 3 • Let and be two segments that except for have no point in common. • Furthermore, let and be two segments that except for have no point in common.
In that case, if • And , • Then .
Axiom 4 • If is an angle • And if is a ray stemming from point • Then there is exactly one ray on a given side of such that • Furthermore, every angle is congruent to itself.
Axiom 5 • If for two triangles and • The congruencies , ,and ∡ are valid (SAS), • Then the congruence ∡ is also satisfied.