Question

Let $\overleftrightarrow{AB}$ be a straight line. Let $\overleftrightarrow{CD}$ and $\overleftrightarrow{EF}$ be two straight lines such that each of them is perpendicular to $\overleftrightarrow{AB}$. Prove that $\overleftrightarrow{CD}\parallel \overleftrightarrow{EF}$.

Answer 1

Let $\overleftrightarrow{AB}$ intersect $\overleftrightarrow{CD}$ and $\overleftrightarrow{EF}$ at L and M respectively.
Since $\overleftrightarrow{CD}\perp \overleftrightarrow{AB}$,
we have $\angle DLA={90}^o$.
Using $\overleftrightarrow{EF}\perp \overleftrightarrow{AB}$,
we also get $\angle FMA={90}^o$.
Thus $\angle DLA=\angle FMA$.
But these are corresponding angles made by the transversal $\overleftrightarrow{AB}$ with the lines $\overleftrightarrow{CD}$ and $\overleftrightarrow{EF}$.
Hence, we conclude that $\overleftrightarrow{CD}\parallel \overleftrightarrow{EF}$.