Chapter 6.3, Problem 19WE

To determine

To prove: that if two lines are perpendicular to the same plane, then the lines do not intersect.

Explanation of Solution

Given information:

Given figure,

  

Calculation:

Consider a plane $P$ such that there are two lines $J$ and $K$ which are perpendicular to the plane $P$ .

Prove that $J$ and $K$ do not intersect as follows:

An indirect proof is initiated assuming temporarily that whatever is need to prove is untrue and then work from there to finally conclude that our assumption is untrue.

Proof:

Assume temporarily that $J$ and $K$ intersect.

However, both $J$ and $K$ are perpendicular to the plane $P$ .

This contradicts the theorem which says that there is exactly one line perpendicular to a given plane through a point outside the plane. In this case are two lines perpendicular to the plane.

Hence, $J$ and $K$ are parallel to each other.

Therefore, $J$ and $K$ do not intersect.