Explain why a line can never intersect a plane in exactly two points. Why can a line never intersect a plane in exactly two points?

I know this seems like a really simple question, but I'm having a hard time figuring out how to answer it. I also tried googling the question but I couldn't find an answer for exactly what I'm looking for.
If you pick two points on a plane and connect them with a straight line then every point on the line will be on the plane. 

Given two points there is only one line passing those points. 

Thus if two points of a line intersect a plane then all points of the line are on the plane.
First, some definitions:


A line is straight and infinitely long.
A plane is flat and infinitely long and wide.
An intersect is a point shared by the line and the plane.


This leads to three relations between a line and a plane:


It is parallel and not part of the plane. This means it has 0 intersects.
It is parallel and part of the plane. This means it has infinite intersects.
It is not parallel. This means at some point it intersects exactly once.


To make a line intersect the same plane twice it can't be parallel, and either the line or the plane must be curved.