It has been taught to us that in order to obtain only the acute or obtuse angle bisectors of two lines, the following algorithm is to be applied:
$($Equation of angle bisectors: ${a_1x + b_1y + c_1 \over \sqrt{(a_1^2 + b_1^2)}} = \pm {a_2x + b_2y + c_2 \over\sqrt{ (a_2^2 + b_2^2)}}$$)$
Make the constant term of the concerned lines positive, i.e., $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ $(c_1,c_2>0)$ are the two lines to be worked with.
Next, check the sign of the product $a_1a_2 + b_1b_2$.
If the product is positive, then the equation of the angle bisector obtained with the positive sign is the obtuse angle bisector, and if the product is negative, then the equation obtained with the negative sign is the equation of the obtuse angle bisector.
I searched rigorously for the proof behind this algorithm but couldn't find anything helpful. Please help me prove this. Any help is appreciated!