Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other
Solution:
Given, two lines are perpendicular to two parallel lines.
We have to show that they are parallel to each other.
Consider two parallel lines m and n and two perpendicular lines p and q.
Given, p ⊥ m and q ⊥ n
So, ∠1 = ∠2 = 90° ---------------------- (1)
We observe that m and n are parallel lines cut by a transversal p.
We know that if two parallel lines are cut by a transversal, then the corresponding angles are equal.
The corresponding angles are ∠1 and ∠3.
So, ∠1 = ∠3
From (1), ∠3 = 90°
Now, ∠1 = ∠2 = ∠3 = 90°
We observe that ∠2 and ∠3 are corresponding angles when the transversal n cuts the lines p and q.
This implies p and q are parallel.
Therefore, two lines which are perpendicular to two parallel lines are parallel to each other.
✦ Try This: In the given figure, if AB∣∣DE, ∠BAC=35° and ∠CDE=53°, find ∠DCE
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 6
NCERT Exemplar Class 9 Maths Exercise 6.3 Problem 10
Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other
Summary:
If a transversal intersects two parallel lines, then corresponding angles are equal and conversely. Two lines are respectively perpendicular to two parallel lines. It is shown that they are parallel to each other
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