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Assume there are no lines through a given point that is parallel to a given line or
assume that there are many lines through a given point that are parallel to a given line.
There exist a line l and a point P not on l such that either there is no line m
parallel to l through P or there are two distinct lines m and n parallel to l through P.
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Does the parallel postulate in Euclidean geometry work in spherical geometry?
No.
Are two lines that are parallel to the same line parallel to each other?
Yes they are. It is delineated in something called the parallel postulate, and the axiom is also called Euclid's fifth postulate. This is boilerplate Euclidean geometry, and a link can be found below if you'd like to review the particulars.
Why hilbert axiom of parallelism assert the existence of only at most one parallel line'?
There is a subtle distinction between Euclidean, Hilbert and Non-Euclidean planes. Euclidean planes are those that satisfy the 5 axioms, while Non-Euclidean planes do not satisfy the fifth postulate. This means that in Non-Euclidean planes, given a line and a point not on that line, then there are two (or more) lines that contain that point and are parallel to the original line. There are geometries where there must be exactly one line through that point and parallel to the original line and then there are also geometries where no such line contains that point and is parallel to the original line.Basically, the fifth postulate can be satisfied by multiple geometries.
What are the names of Non-Euclidean Geometries?
Answer The two commonly mentioned non-Euclidean geometries are hyperbolic geometry and elliptic geometry. If one takes "non-Euclidean geometry" to mean a geometry satisfying all of Euclid's postulates but the parallel postulate, these are the two possible geometries.
What is the difference between Euclidean Geometry and non-Euclidean Geometry?
In Euclidean geometry parallel lines are always the same distance apart. In non-Euclidean geometry parallel lines are not what we think of a parallel. They curve away from or toward each other. Said another way, in Euclidean geometry parallel lines can never cross. In non-Euclidean geometry they can.
