Question

State true or false
Two lines that are respectively perpendicular to two intersecting lines always intersect each other.

Answer 1

We prove this statement is true or false with the help of a diagram and with help of concepts of perpendicular lines, parallel lines and intersecting lines.

• Perpendicular means at that point angle is a right angle
• Parallel lines never intersect each other.

Complete step-by-step answer:
We draw a diagram for the solution

Here two lines $m,n$ are intersecting at a point $O$
And we draw perpendiculars to each line, $PQ \bot n,RS \bot m$
Statement in the question says that two lines that are respectively perpendicular to two intersecting lines always intersect each other.
In this case we have to find if the two perpendiculars $PQ,RS$intersect each other or not.
Let us assume that two perpendiculars never intersect.
So, $PQ,RS$ will never meet at any point which means that lines $PQ,RS$are parallel to each other.
Now we know $PQ \bot n$ and $PQ\parallel RS$, so the perpendicular will pass the parallel line $RS$at a right angle so, $PQ \bot n$. So $n$ acts as a common perpendicular for the set of parallel lines $PQ,RS$
Similarly, we know $RS \bot m$ and $PQ\parallel RS$, so the perpendicular will pass the parallel line $PQ$at a right angle so, $PQ \bot m$. So $m$ acts as a common perpendicular for the set of parallel lines $PQ,RS$
Since, $m,n$ both become perpendicular to the parallel lines $PQ,RS$means that $m,n$will also be parallel which is a contradiction to our assumption.
Therefore, two perpendiculars to the intersecting lines will intersect each other. The statement is TRUE

Note: Students might get confused with the fact that the perpendicular can be taken on the opposite side of the line then it won’t intersect but they should keep in mind that perpendicular lines can be extended in both the directions and they will intersect eventually.