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Question

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

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Solution

Given, two lines m and n are parallel and another two lines p and q are respectively perpendicular to m and n.
i.e., pm, pn, qm, qn,
To prove p || q,
Proof:
m || n and p are perpendicular to m and n.

1=10=90 [corresponding angles]
Similarly, 2=9=90 [corresponding angles]
[ p m and p n]
4=10=90 and 3=9=90.........(i) [alternate interior angles]
Similarly, if m || n and q is perpendicular to m and n,
Then, 7=90 and 11=90.....(ii)
From (i) and (ii)
Now, 4+7=90+90=180
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.

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