Question

prove that lines which are parallel to the same line are parallel to each other​

Answer 1

Answer:

Consider a line l, and suppose that we draw two more lines m and n such that l ∥∥ m and l ∥∥ n.



Can we say that m ∥∥ n? Draw any transversal across the three lines, as we have done above, and note that

∠1∠1 = ∠2∠2 (corresponding angles)

∠2∠2 = ∠3∠3 (corresponding angles)

Thus,

∠1∠1 = ∠3∠3

This proves that m must be parallel to n. We formalize this result in the form of a theorem.

Theorem: Two or more lines which are parallel to the same line will be parallel to each other.

Consider a line l, and consider two more lines mand n such that l ∥∥ m and l ∥∥ n, as shown below: 



The distance between l and m is x, and the distance between l and n is y. What is the distance between m and n? The theorem above tells us that m and n will also be parallel, and therefore there will be a fixed (constant) distance between them. Clearly, that distance will be x + y.