Prove that the two lines which are both parallel to the same line are parallel to one another.

Explanation

GIVEN Three lines `l, m, n` such that `l||n and m||n`.
TO PROVE `l||m`.
CONSTRUCTION Draw a transversal t, cutting `l, m, n` at `E, F, G` respectively.
PROOF Since `l||n` and transversal t cuts them at E and G respectively, so
`angle1=angle3` (corres. `angles`).
Again `m||n` and transversal t cuts them at F and G respectively, so
`angle2=angle3` (corres. `angles`).
`:. angle1=angle2` (each equal to `angle3`).
But these are corresponding angles formed when the transversal t cuts `l` and `m` at E and F respectively.
`:. l||m` (by corres. `angles` axiom).