1. ## Parallel Proof

Prove that nonvertical lines are parallel if and only if their slopes are equal.

2. Lines can be written in the form $\displaystyle \displaystyle y = mx + c$, where $\displaystyle \displaystyle m$ is the gradient and $\displaystyle \displaystyle c$ is the $\displaystyle \displaystyle y$ intercept.

So we could write the equation of two lines as

$\displaystyle \displaystyle y=m_1x + c_1$ and $\displaystyle \displaystyle y=m_2x + c_2$.

You should know that parallel lines never cross. These two lines will cross when their equations are equal. So

$\displaystyle \displaystyle m_1x + c_1 = m_2x + c_2$

$\displaystyle \displaystyle m_1x - m_2x = c_2 - c_1$

$\displaystyle \displaystyle x(m_1-m_2) = c_2 - c_1$

$\displaystyle \displaystyle x = \frac{c_2 - c_1}{m_1-m_2}$.

Obviously there is no solution when the denominator is $\displaystyle \displaystyle 0$, in other words, where $\displaystyle \displaystyle m_1 = m_2$.

So the only time when there will not be a point of intersection is when the two gradients are equal.

In other words, the lines are parallel when their gradients are equal.