Question

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. True
• B. False

Answer 1

The correct option is A True

Take any line $l$ and a point $P$ not on $l$. Then by play Fair'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 a point from a line is the length of the perpendicular from the point to the line. This distance will be the same for any point on $m$ from $l$ and any point on $l$ from $m$. Thus these two lines are everywhere equidistance from one another.