For every line L and for every point P not lying on a given line L, there exists _____________ passing through P and parallel to L.

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No line

A unique line

Two lines

Infinitely many lines

answer is B.

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Detailed Solution

Concept- The equidistance postulate, Playfair's axiom, Proclus' axiom, the triangle postulate, and the Pythagorean theorem are all equivalent to the parallel postulate. In Hilbert's axioms, there is also a single parallel axiom that is equivalent to Euclid's parallel postulate.
Playfair's axiom is indeed a geometric axiom that can be used instead of Euclid's fifth postulate (the parallel postulate):
If there is a line and a point not on it in a plane, only one line parallel to the given line can be drawn through the point. It doesn’t matter how far they are extended.
Hence, the correct answer is option 2) a unique line.

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