Chapter 6.3, Problem 10ST1

To determine

To prove: There is at most one line through $P$ perpendicular to $k$

Explanation of Solution

Given information: Point $P$ is not on the line $k$ .

Proof:

  

Steps have been arranged as follows:

(d) Assume temporarily that there are two lines through P perpendicular to k at A and B.

(b) Then $\angle PAB$ and $\angle PBA$ are right angles, and $\Delta PAB$ has two right angles.

(a) But this contradicts Corollary 3 Of Theorem 3-1: In a triangle, there can be at most one right angle or obtuse angle.

(c) Thus, our temporary assumption is false, and there is at most one line through $P$ perpendicular to $k$ .