Hilbert’s Axioms for Euclidean GeometryAxioms of Congruence

JessicaErica

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 20th century.

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 .

𝐵 ′

𝐴 𝐵𝑎

𝐴 ′ 𝑎 ′

𝐴 ′ 𝑎 ′

𝐴 𝑎 𝐵

𝐵 ′

Axiom 2If

And if ,

Then

𝐴 ′ 𝐵 ′

𝐴 𝐵𝐴 ′ ′ 𝐵 ′ ′

𝐴 𝐵

𝐴 ′ 𝐵 ′

𝐴 ′ ′ 𝐵 ′ ′

Axiom 3Let 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.

𝐴 𝐵 𝐶

𝐴 ′ 𝐵 ′ 𝐶 ′

In that case, if And ,

Then .

𝐴 𝐵 𝐶

𝐴 ′ 𝐵 ′ 𝐶 ′

𝐴 𝐵 𝐶

𝐴 ′ 𝐵 ′ 𝐶 ′

Axiom 4If is an angleAnd 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.

𝐴

𝐵 𝐶

𝐵 ′ 𝐶 ′

𝐴 ′

Axiom 5If for two

triangles and

The congruencies , , and ∡ are valid (SAS),

Then the congruence ∡ is also satisfied.

𝐴 𝐵

𝐶

𝐴 ′

𝐶 ′

𝐵 ′