Question

Question 10
Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.

Answer 1

Given, two lines m and n are parallel and another two lines p and q are respectively perpendicular to m and n.
$i.e.,p\perp m,p\perp n,q\perp m,q\perp n,$
To prove p || q,
Proof:
m || n and p are perpendicular to m and n.

$\therefore \angle 1=\angle 10={90}^{\circ }$ [corresponding angles]
Similarly, $\angle 2=\angle 9={90}^{\circ }$ [corresponding angles]
[$\because$ p $\perp$ m and p $\perp$ n]
$\therefore \angle 4=\angle 10={90}^{\circ }and\angle 3=\angle 9={90}^{\circ }$.........(i) [alternate interior angles]
Similarly, if m || n and q is perpendicular to m and n,
Then, $\angle 7={90}^{\circ }and\angle 11={90}^{\circ }$.....(ii)
From (i) and (ii)
Now, $\angle 4+\angle 7={90}^{\circ }+{90}^{\circ }={180}^{\circ }$
So, sum of two interior angles is supplementary.
We know that, if a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.
Hence, p || q.