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Hilbert’s Axioms for Euclidean Geometry Axioms of Congruence

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 Geometry Axioms of Congruence

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  1. 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.

  2. 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.

  3. 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 .

  4. Axiom 2 • If • And if , • Then

  5. 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.

  6. In that case, if • And , • Then .

  7. 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.

  8. Axiom 5 • If for two triangles and • The congruencies , ,and ∡ are valid (SAS), • Then the congruence ∡ is also satisfied.

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