Given points $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$, and $D(x_4,y_4)$, we can form vectors $\vec{AB}(x_2-x_1,y_2-y_1)$ and $\vec{CD}(x_4-x_3,y_4-y_3)$. Then the cosine of an angle $\alpha$ between vectors $\vec{AB}$ and $\vec{CD}$ will be equal to:
$$ \cos\alpha = \dfrac{(x_2-x_1)(x_4-x_3) + (y_2-y_1)(y_4-y_3)}{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\sqrt{(x_4-x_3)^2+(y_4-y_3)^2}} $$
Are there any special cases which would allow to simplify the calculation of this formula? I'm looking for a list of as many of these cases as possible.