Question
Prove that two lines that are respectively perpendicular to two intersecting lines intersect each other. [Hint: Use proof by contradiction].
✓
Answer
Given: Let lines $l$ and $m$ are two intersecting lines. Again, let n and p be another two lines which are perpendicular to the intersecting lines meet at point $D.$
To prove: Two lines $n$ and $p$ intersecting at a point.
Proof: Let us consider lines $n$ and $p$ are not intersecting, then it means they are parallel to each other i.e., $n || p.....(1)$ Since, lines $n$ and $p$ are perpendicular to $m$ and $l$ respectively.
But from equation $(1), n || p,$ it implies that $l$ and $m.$ It is a contradiction.
Thus, our assumption is wrong.
Hence, lines $n$ and $p$ intersect at a point.
To prove: Two lines $n$ and $p$ intersecting at a point.
Proof: Let us consider lines $n$ and $p$ are not intersecting, then it means they are parallel to each other i.e., $n || p.....(1)$ Since, lines $n$ and $p$ are perpendicular to $m$ and $l$ respectively.
But from equation $(1), n || p,$ it implies that $l$ and $m.$ It is a contradiction.
Thus, our assumption is wrong.
Hence, lines $n$ and $p$ intersect at a point.
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