# Parallel Straight Lines have Same Slope

## Theorem

Let $L_1$ and $L_2$ be straight lines in the plane.

Let $L_1$ and $L_2$ have slopes of $m_1$ and $m_2$ respectively.

Then $L_1$ and $L_2$ are parallel if and only if $m_1 = m_2$.

## Proof

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

 $\displaystyle L_1: \ \$ $\displaystyle y$ $=$ $\displaystyle m_1 x + c_1$ $\displaystyle L_2: \ \$ $\displaystyle y$ $=$ $\displaystyle m_2 x + c_2$

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

 $\displaystyle \tan \psi_1$ $=$ $\displaystyle m_1$ $\displaystyle \tan \psi_2$ $=$ $\displaystyle m_2$

### 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.

$\Box$

### Sufficient Condition

Suppose $L_1 \parallel L_2$.

Then:

 $\displaystyle \phi_1$ $=$ $\displaystyle \phi_2$ Parallelism implies Equal Corresponding Angles $\displaystyle \leadsto \ \$ $\displaystyle \tan \phi_1$ $=$ $\displaystyle \tan \phi_2$ $\displaystyle \leadsto \ \$ $\displaystyle m_1$ $=$ $\displaystyle m_2$

$\blacksquare$