Consider the following statement: There exists a pair of straight lines that are everywhere equidistant from one another. Is this statement a direct consequence of Euclid's fifth postulate? Explain.
(A) Take any line $l$ and a point $P$ not on $l$.
According to Playfair's axiom,which is equivalent to Euclid's fifth postulate,there exists a unique line $m$ passing through $P$ that is parallel to $l$.
By definition,the distance of a point from a line is the length of the perpendicular segment drawn from the point to the line.
Since the lines are parallel,the perpendicular distance from any point on line $m$ to line $l$ remains constant.
Therefore,these two lines are everywhere equidistant from one another,which is a direct consequence of the fifth postulate.
According to Playfair's axiom,which is equivalent to Euclid's fifth postulate,there exists a unique line $m$ passing through $P$ that is parallel to $l$.
By definition,the distance of a point from a line is the length of the perpendicular segment drawn from the point to the line.
Since the lines are parallel,the perpendicular distance from any point on line $m$ to line $l$ remains constant.
Therefore,these two lines are everywhere equidistant from one another,which is a direct consequence of the fifth postulate.
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