Write an indirect proof to prove theorem 4-2: if there is a line and a point not on a line, then exactly one plane contains them.

Question

Please write the indirect proof to solve this.

Answer 1

Given a two separate points p₁ and p₂ in space we know there is always a line k passing through those two points, Given a two separate lines q₁ and q₂ there is always a plane r passing through those two lines.

for the problem above let there be line t and point s such that point s is not on the line t. we know there exist a plane g that passes through both line t and point s. There could be no other plane that passes through both line t and point s because s cannot be on two different planes simultaneously unless it is on the line t. if it was the case that line t and s are intersected by more than one plane, line t and point s would be on the line of intersection of the planes. Hence, if there is line such and a point not on a line, then there is exactly a one plane that contains both.