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Take a line l and a point P not on l. Then, by Playfair's axiom, which is equivalent to the fifth postulate, we know that there is a unique line m through P which is parallel to l.
Now, the distance of the point from line is the length of the perpendicular from the line. This, distance will be the same for any point on m from l and any point on l from m. So, these two lines are everywhere equidistant from one another.
Now, the distance of the point from line is the length of the perpendicular from the line. This, distance will be the same for any point on m from l and any point on l from m. So, these two lines are everywhere equidistant from one another.
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