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This postulate does not specifically talk about parallel lines; [1] it is only a postulate related to parallelism. 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 Upper and Lower Dimension axioms together require that any model of these axioms have dimension 2, i.e. that we are axiomatizing the Euclidean plane. Suitable changes in these axioms yield axiom sets for Euclidean geometry for dimensions 0, 1, and greater than 2 (Tarski and Givant 1999: Axioms 8 (1), 8 (n), 9 (0), 9 (1), 9 (n)).
A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
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.
[1] 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.
Saccheri quadrilaterals. A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base.It is named after Giovanni Gerolamo Saccheri, who used it extensively in his 1733 book Euclides ab omni naevo vindicatus (Euclid freed of every flaw), an attempt to prove the parallel postulate using the method reductio ad absurdum.
[1] It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry [2] and was named after the Scottish mathematician John Playfair. The "at most" clause is all that is needed since it can be proved from the first four axioms that at least one parallel line exists given a line L and a point P not on L, as follows:
[3] Since Euclid's postulate is equivalent to the statement that the sum of the internal angles of a triangle is 180°, he considered both the hypothesis that the angles add up to more or less than 180°. The first led to the conclusion that straight lines are finite, contradicting Euclid's second postulate. So Saccheri correctly rejected it.