Prove that two lines that are respectively perpendicular to two intersecting lines intersect each other.

[Hint: Use proof by contradiction]

Let us draw the figure as below –

It is given to us that

Two lines are intersecting each other. Let us assume l and m to be the two intersecting lines.

Also, we have two lines that are perpendicular to the two intersecting lines. Let us say, a ⊥ l, and b ⊥ m.

We have to prove that a and b intersect each other.

Let us assume that a and b do not intersect.

⇒ a || b

Now, we have a ⊥ l and, a || b

⇒ b ⊥ l - - - - (i)

Also, we have b ⊥ m - - - - (ii)

From (i) and (ii), we can say that this situation will hold true if and only if l || m.

But, this is incorrect because it is given to us that l and m are two intersecting lines.

Hence, our initial assumption is wrong.

Thus, a and b intersect each other.

Therefore, it is proved that two lines that are respectively perpendicular to two intersecting lines intersect each other.