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If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side. 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.
The various attempted proofs of the parallel postulate produced a long list of theorems that are equivalent to the parallel postulate. Equivalence here means that in the presence of the other axioms of the geometry each of these theorems can be assumed to be true and the parallel postulate can be proved from this altered set of axioms.
With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects beyond the proof-theorem framework, [27] in experimental mathematics. Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development of fractal ...
Parallel lines are the subject of Euclid's parallel postulate. [2] Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry , lines can have analogous properties that are referred to as parallelism.
Five color theorem; Five lemma; Fundamental theorem of arithmetic; Gauss–Markov theorem (brief pointer to proof) Gödel's incompleteness theorem. Gödel's first incompleteness theorem; Gödel's second incompleteness theorem; Goodstein's theorem; Green's theorem (to do) Green's theorem when D is a simple region; Heine–Borel theorem ...
Analyst's traveling salesman theorem (discrete mathematics) Analytic Fredholm theorem (functional analysis) Anderson's theorem (real analysis) Andreotti–Frankel theorem (algebraic geometry) Angle bisector theorem (Euclidean geometry) Ankeny–Artin–Chowla theorem (number theory) Anne's theorem ; Apéry's theorem (number theory)
Historically, Euclid's parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry. Other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry.
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels.