Two Non-Parallel Lines Intersect Exactly Once

An interesting way to look at this idea is to place the two lines in an arbitrarily-oriented Cartesian coordinate plane. Then we can express the lines with the equations $y=m_1x+b_1$ and $y=m_2x+b_2$. Now, because the lines are not parallel, it follows that $m_1 \neq m_2.$ We want to show that there exists some $(x,y)$ that satisfies both equations. We see that $$(x,y) = \left( \frac{b_1-b_2}{m_2-m_1}, \frac{b_1m_2-b_2m_1}{m_2-m_1} \right)$$ is indeed a solution, so there must exists an intersection between these two lines.

Two non-parallel lines intersect. The distance between them starts out very long, then reduces until it is 0 at the intersection point. Once again, it increases forever, but gradually.

So, the definition of parallel is lines that are coplanar and will never intersect. So, if two lines are in the same plane and are not parallel, its the opposite of the definition, right? Therefore, the lines are not parallel.

So if the lines weren't in the same plane, this rule wouldn't apply, unless the planes intersected. If not, then the lines are skew.