Question
Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.
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Answer
Let the two parallel lines be $m$ and $n$.
Let $p ⊥ m$. $\Rightarrow\angle1=90^\circ$
Let q ⊥ n. $\Rightarrow\angle2=90^\circ$
Now, m || n and p is a transversal.
$\Rightarrow\angle1=\angle3$ (corresponding angles)
$\Rightarrow\angle3=90^\circ$ $\Rightarrow\angle3=\angle2$ $($each $90^\circ)$
But, these are corresponding angles, when transversal $n$ cuts lines $p$ and $q$.
$\therefore$ $p || q$.
Hence, two lines which are perpendicular to two parallel lines, are parallel to each other.
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