Angle Between Lines, Parallelism, Perpendicularity and Distance Formulas

Once two lines are written with known slopes $m_1$ and $m_2$, several useful facts follow directly.

Angle between two lines. If $\theta$ is the acute angle between two lines of slopes $m_1$ and $m_2$:

Parallel lines have equal slopes: $m_1 = m_2$. In general form, $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ are parallel exactly when $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2}$.

Perpendicular lines have slopes whose product is $-1$: $m_1 m_2 = -1$. (When one line is vertical, the other must be horizontal for the pair to be perpendicular.)

Distance of a point $(x_0,y_0)$ from a line $ax+by+c=0$:

Distance between two parallel lines $ax+by+c_1=0$ and $ax+by+c_2=0$ (same $a,b$ after scaling both to the same coefficients):

$d = \dfrac{|c_1-c_2|}{\sqrt{a^2+b^2}}$ …