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