Question

Prove : Two lines which are parallel to a common line are parallel to each other.

Answer 1

Suppose $\overleftrightarrow{AB}$ and $\overleftrightarrow{XY}$ are two lines which are parallel to a common line $\overleftrightarrow{CD}$.
We show that $\overleftrightarrow{AB}\parallel \overleftrightarrow{XY}$ . Draw a transversal $\overleftrightarrow{APQ}$, cutting $\overleftrightarrow{AB}$ at $L$, $\overleftrightarrow{CD}$ at $M$ and $\overleftrightarrow{XY}$ at $N$, respectively.
We observe that $\angle BLP=\angle DMP$, as they are corresponding angles made by transversal $\overleftrightarrow{PQ}$ with the parallel lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$.

Similarly, $\angle DMP=\angle YNP$

Using Axiom 1, we obtain $\angle BLP=\angle YNP$.
Hence, we can conclude that $\overleftrightarrow{AB}\parallel \overleftrightarrow{XY}$.