Two lines are respectively perpendicular to two parallel lines.

Question:

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

Solution:

Let the two lines m and n be respectively perpendicular to the two parallel lines p and q.
To prove: is parallel to n.
Proof: Since, m is perpendicular to p

$\therefore \angle 1=90^{\circ}$

Also, ​n is perpendicular to 

$\therefore \angle 3=90^{\circ}$

Since p and are parallel and is a transversal line 

$\therefore \angle 2=\angle 1=90^{\circ} \quad$ [Corresponding angles]

Also, $\angle 2=\angle 3=90^{\circ}$

We know that if two corresponding angles are equal then the two lines containing them must be parallel.
Therefore, the lines m and n are parallel to each other.

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