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Playfair's axiom states:

Through any point in the plane, there is at most one straight line parallel to a given straight line.

This axiom is equivalent to the parallel postulate.

If the point $P$ lies on the given straight line $\mathcal{l}$, our intuition tells us that there are no straight lines through $P$ that are parallel to $\mathcal{l}$. This is because any straight line passing through $P$ either intersects $\mathcal{l}$ at one point, namely $P$, (which means they are not parallel) or is $\mathcal{l}$ itself (a line cannot be parallel to itself).

This seems obvious to me but have I implicitly used Playfair's axiom or any of the first four postulates?

Siddhartha
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    The definition of "parallel" in Euclidean geometry is "non-intersecting", i.e. having no common points. You do not need any geometric postulates to assert that two lines with a common point do have common points, only pure logic. – Conifold Dec 27 '19 at 12:00
  • @Conifold I think you're right. Thanks! – Siddhartha Dec 27 '19 at 12:03

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