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

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

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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 \parallel q

Proof:

Since m||n and p is perpendicular to m and n.
\angle 5 = \angle 6 = \angle 7 = \angle 8 = \angle 11 = \angle 12 = 90^{\circ}
Similarly, q is perpendicular to m and n.
Ð\\angle 1 = \angle 2 = \angle 3 = \angle 4 = \angle 9 = \angle 10 = 90^{\circ}
Now for lines p and q, m is the transversal
\angle 1 = \angle 2 = \angle 3 = \angle 4 = \angle 5 = \angle 6 = \angle 7 = \angle 8 = 90^{\circ}
So we can see that all the conditions are fulfilled for the lines to be parallel, i.e., Corresponding angles are equal, sum of cointerior angles is 180o, alternate angles are equal.

Hence, p \parallel q

Hence proved

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