Question

Prove that two lines which are both 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.

Corresponding angles

Can we say that m

∥

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

∠

1

=

∠

2

(corresponding angles)

∠

2

=

∠

3

(corresponding angles)

Thus,

∠

1

=

∠

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 m and n such that l

∥

m and l

∥

n, as shown below:

Constant distance between parallel lines

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.