Condition for Straight Lines in Plane to be Parallel/Slope Form/Proof 2

Proof
Let $L_1$ and $L_2$ be embedded in a cartesian plane, given by the equations:

Let $\phi_1$ and $\phi_2$ be the angles that $L_1$ and $L_2$ make with the $x$-axis respectively.

Then by definition of slope of a straight line:

Necessary Condition
Let $m_1 = m_2$.

Then:
 * $\tan \psi_1 = \tan \psi_2$

and so:
 * $\psi_1 = \psi_2 + n \pi$

The multiple of $\pi$ makes no difference.

Thus from Equal Corresponding Angles implies Parallel Lines, $L_1$ and $L_2$ are parallel.

Sufficient Condition
Suppose $L_1 \parallel L_2$.

Then: