Question

AB is a given line. Draw XY and PQ
parallel to AB on either side of AB. Check weather XY and PQ are also parallel to each other.​

Answer 1

Hypothesis: XY || PQ and PQ ∥ XY

Case 1: XY = PQ. By convention, XY ∥ PQ.

Case 2: XY is not equal to PQ.

Let us assume that XY is not parallel to PQ

Then XY and PQ intersect in at least one point m, which implies that XY and PQ are distinct lines parallel to AB passing through m.

This contradicts the Parallel Postulate that there exists only one line parallel to AB passing through m.

Therefore, our assumption that is XY is not parallel to PQ is false, so we conclude that XY || PQ.