Question

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 ⊥ m, p ⊥ n, q ⊥ m, q ⊥ n

To prove p || g

Proof Since, m || n and p is perpendicular to m and n.

∴ ∠1 = ∠10 = 90°  ...[Corresponding angles]

Similarly, ∠2 = ∠9 = 90°  ...[Corresponding angles]

∴ ∠4 = ∠9 = 90° and ∠3 = ∠10 = 90°  ...[Alternative interior angles] [∵ p ⊥ m and p ⊥ n]

 Similarly, if m || n and q is perpendicular to m and n.

Then, ∠7 = 90° and ∠11 = 90°

Now, ∠3 + ∠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 || g.