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Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements.Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions from these.
It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics.
Euclid gave the definition of parallel lines in Book I, Definition 23 [2] just before the five postulates. [3] Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. The postulate was long considered to be obvious or inevitable, but proofs were elusive.
Absolute geometry is a geometry based on an axiom system consisting of all the axioms giving Euclidean geometry except for the parallel postulate or any of its alternatives. [69] The term was introduced by János Bolyai in 1832. [70] It is sometimes referred to as neutral geometry, [71] as it is neutral with respect to the parallel postulate.
The axiomatic foundation of Euclidean geometry can be dated back to the books known as Euclid's Elements (circa 300 B.C.). These five initial axioms (called postulates by the ancient Greeks) are not sufficient to establish Euclidean geometry. Many mathematicians have produced complete sets of axioms which do establish Euclidean geometry.
A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions. One must concede the need for primitive notions, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical ...
In Euclidean geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent. The AA postulate follows from the fact that the sum of the interior angles of a triangle is always equal to 180°. By knowing two angles, such as 32° and 64° degrees, we know that the next angle is 84°, because 180 ...
Supporting hyperplane theorem (convex geometry) Swan's theorem (module theory) Sylow theorems (group theory) Sylvester's determinant theorem (determinants) Sylvester's theorem (number theory) Sylvester pentahedral theorem (invariant theory) Sylvester's law of inertia (quadratic forms) Sylvester–Gallai theorem (plane geometry)
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