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The hyperbolic parallel postulate states that given a line L and a point P, not on the line, there are at least two distinct lines through P that do not intersect L.

The negation is that given a line L and a point P, not on the line, there is at most one line through P that does not intersect L.

The negation includes the case where there is exactly one such line - which is the Euclidean space.

Q: How to negate the hyperbolic parallel postulate?

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postulate theorems tell that the lines are parallel, but the converse if asking you to find if the lines are parallel.

euclidean Geometry where the parallel line postulate exists. and the is also eliptic geometry where the parallel line postulate does not exist.

This is Euclid's fifth postulate, also known as the Parallel Postulate. It is quite possible to construct consistent systems of geometry where this postulate is negated - either many parallel lines or none.

converse of the corresponding angles postulate

It is a consequence of Euclid's parallel postulate. In fact, in some versions, the statement that "a plane triangle has interior angles that sum to 180 degrees" replaces the parallel postulate.

Related questions

Hyperbolic geometry is a beautiful example of non-Euclidean geometry. One feature of Euclidean geometry is the parallel postulate. This says that give a line and a point not on that line, there is exactly one line going through the point which is parallel to the line. (That is to say, that does NOT intersect the line) This does not hold in the hyperbolic plane where we can have many lines through a point parallel to a line. But then we must wonder, what do lines look like in the hyperbolic plane? Lines in the hyperbolic plane will either appear as lines perpendicular to the edge of the half-plane or as circles whose centers lie on the edge of the half-plane

by creating two planes such that one parallel is hyperbolic and the other parabolic

postulate theorems tell that the lines are parallel, but the converse if asking you to find if the lines are parallel.

euclidean Geometry where the parallel line postulate exists. and the is also eliptic geometry where the parallel line postulate does not exist.

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.

This is Euclid's fifth postulate, also known as the Parallel Postulate. It is quite possible to construct consistent systems of geometry where this postulate is negated - either many parallel lines or none.

Parallel lines are parallel. Proof they have same slopes

converse of the corresponding angles postulate

Parallel Postulate

Playfair Axiom

parallel postulate

... given line. This is one version of Euclid's fifth postulate, also known as the Parallel Postulate. It is quite possible to construct consistent systems of geometry where this postulate is negated - either many parallel lines or none.