Properties of two parallel lines cut by a transversal
Two lines are...
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.,p⊥m,p⊥n,q⊥m,q⊥n,
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.