Question
Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.
✓
Answer
Two lines $p$ and $n$ are respectively perpendicular to two parallel line $l$ and $m$, i.e., $\text{p}\perp\text{l}$ and $\text{n}\perp\text{m}$
We have to show that $p$ is parallel to $n.$
As $\text{n}\perp\text{m},$ So $\angle1=90^\circ....(1)$
Again, $\text{p}\perp\text{l},$ So $\angle2=90^\circ.$
But, $l$ is parallel to $m,$ so
$\angle1=\angle3$ $[\text{corres}.\angle\text{s}]$
$\therefore\angle2=\angle90^\circ...(2)$ $[\because\angle2=90^\circ]$
From $(1)$ and $(2),$ we get
$\Rightarrow\angle1=\angle3$ $[\text{Each}=90^\circ]$
angles.
Hence, $p||n.$
But, these are corresponding.
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