absolute geometry Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates, but since these are not suf ...

, the Saccheri–Legendre theorem states that the sum of the angles in a

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

is at most 180°. Absolute geometry is the geometry obtained from assuming all the

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

s that lead to

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

with the exception of the axiom that is equivalent to the

parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...

of Euclid.There are many axiom systems that yield Euclidean geometry and they all contain an axiom that is logically equivalent to Euclid's parallel postulate. The theorem is named after

Giovanni Girolamo Saccheri Giovanni Girolamo Saccheri (; 5 September 1667 – 25 October 1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. Saccheri was born in Sanremo. He entered the Jesuit order in 1685 and was ordained as a priest in 1694 ...

and

Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...

. The existence of at least one triangle with angle sum of 180 degrees in absolute geometry implies Euclid's parallel postulate. Similarly, the existence of at least one triangle with angle sum of less than 180 degrees implies the characteristic postulate of

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...

Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...

gave an example of a

non-Legendrian geometry In geometry, Max Dehn introduced two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel to a given one that pass through a given point, but where the sum of the angles of a triangle ...

where the angle sum of a triangle is greater than 180 degrees, and a

semi-Euclidean geometry In geometry, Max Dehn introduced two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel to a given one that pass through a given point, but where the sum of the angles of a triangle ...

where there is a triangle with an angle sum of 180 degrees but Euclid's parallel postulate fails. In Dehn's geometries the

Archimedean axiom In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typical ...

does not hold.

Notes

Euclidean geometry Theorems about triangles Non-Euclidean geometry {{geometry-stub