How do I prove this method of determining the sign for acute or obtuse angle bisector in the angle bisector formula works? The formula for finding the angular bisectors of two lines $ax+by+c=0$ and $px+qy+r=0$ is $$\frac{ax+by+c}{\sqrt{a^2+b^2}} = \pm\frac{px+qy+r}{\sqrt{p^2+q^2}}$$
I understand the proof of this formula but I do not understand how to determine which sign is for acute bisector and which one for obtuse.
I can find the angle between a bisector and a line, and if it comes less than $45^\circ$ then it is acute bisector.
But that is a lengthy method and involves calculation.
My book says, if $ap+bq$ is positive then the negative sign in the formula is for acute bisector.
I want a proof of this method.
Edit: Using the method for finding the position of two points with respect to a line is okay for the proof.

If you follow the method described by me here, we simply have to check the angle between normal after finding the bisector vectors:

Now, check the angle between $n_1$ and $n_2$, if it is greater than ninty degrees then it means $\chi$ is the obtuse bisector and if not, it $\chi$ is the acute bisector. To understand why, check the linked in equi-angle form.

