Define Euclid's 5th postulate ?
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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 the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
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 postulate was long considered to be obvious or inevitable, but proofs were elusive. Eventually it was discovered that inverting the postulate gave valid, albeit different geometries. A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or sometimes "neutral geometry"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 the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
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 postulate was long considered to be obvious or inevitable, but proofs were elusive. Eventually it was discovered that inverting the postulate gave valid, albeit different geometries. A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or sometimes "neutral geometry"
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