Question

prove that the line segments joining the same side of equal and parallel line are also equal and parallel to themselves

Answer 1

Answer:

Well, it seems appropriate to call two segments parallel if their containing lines are parallel. Since we’re given that these two segments belong to the same line, the question then becomes: Is a line parallel to itself?

If we go by The Book (and here I mean Euclid’s Elements, which was used as a textbook for over two millennia), the answer is No. That is because, according to Euclid’s Definition 23:

“Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.”

…and obviously a line meets itself (and in fact does so an infinite number of times).

That said, and as many have observed, what we’re discussing here is just a terminological convention, and one can make a pretty strong case for the alternative convention that a line *is* considered parallel to itself, because this a) allows the question of parallelism to be answered by simply checking whether the slopes are equal, and b) makes “is parallel to” into an equivalence relation, a mathematical property which is useful to know in some contexts.

My summary take: If you’re an originalist/traditionalist, No. If you like the two properties mentioned in the previous paragraph, Yes (but be clear about this when the distinction matters). If you don’t want to waste brain cells deciding, then just use the convention of your community/culture/textbook. And if even *that’s* too hard to sort out, go with the Old Guy: No. (It’s hard to beat twenty centuries on the bestseller list.