Question

Please explain me playfair's axiom.......and postulate 5 of Euclid proposed by him

Answer 1

In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

Euclid's fifth postulate implies Playfair's axiom

The easiest way to show this is using the Euclidean theorem (equivalent to the fifth postulate) that states that the angles of a triangle sum to two right angles. Given a line ℓ and a point P not on that line, construct a line, t, perpendicular to the given one through the point P, and then a perpendicular to this perpendicular at the point P. This line is parallel because it cannot meet ℓ and form a triangle, which is stated in Book 1 Proposition 27 in Euclid's Elements.Now it can be seen that no other parallels exist. If n was a second line through P, then n makes an acute angle with t (since it is not the perpendicular) and the hypothesis of the fifth postulate holds, and so, n meets ℓ

Playfair's axiom implies Euclid's fifth postulate
Given that Playfair's postulate implies that only the perpendicular to the perpendicular is a parallel, the lines of the Euclid construction will have to cut each other in a point. It is also necessary to prove that they will do it in the side where the angles sum to less than two right angles, but this is more difficult.