Let $ax^2+2hxy+by^2=0$ be the equation of two straight lines passing through the origin. We know that the angle between these two straight lines is given by, $$\arctan \dfrac{2\sqrt{h^2-ab}}{a+b}$$
The condition for perpendicularity is given by $$a+b=0$$
Is it not illogical? Since we already know (and in fact can prove) that division by $0$ is undefined, how can we define the $\arctan$ of an undefined quantity?
I think this statement is sloppy, so you are making a good point. What is should be said is that either the lines are perpendicular, for which the test is
$a+b=0$,
or the lines are not perpendicular, in which case the smaller of the two angles made by the two intersecting lines is given by the formula you give.