Transitivity Of Parallelism

Theorem: Two lines, each parallel to a third line, are parallel to each other.

Prerequisites:
Parallel Lines (definition)

Proof:

Let the lines $l_1$ and $l_3$ be parallel to $l_2$ in the given figure. Without any loss of generality, we can bring $l_2$ in between $l_1$ and $l_3$.

Now, if possible, let us assume that the lines $l_1$ and $l_3$ are not parallel to each other. Hence, by definition of parallel lines, $l_1$ and $l_3$ must intersect each other. For intersection, at least on of the line $l_1$ or $l_3$ must cross the line $l_2$, as $l_2$ is in between $l_1$ and $l_3$. But this contradicts the given condition that $l_2$ is parallel to both $l_1$ and $l_3$.

Hence, $l_1$ and $l_3$ cannot intersect, and are therefore parallel.


Recommended:
Unique parallel through a point
Corresponding  angles property
Alternate angles property