What is the logic behind using tangent lines to find the angle of intersection of two curves?
I can understand why you need to find the point of intersection, and I can understand why you use the dot product to find the angle, but I don't understand why they used tangent lines to $r1$ $r2$ respectively?
If you'll forgive my putting words in your mouth: To define (and measure) the angle of intersection between two curves at a point $p$ where the curves intersect, we should look at $p$ Really Closely, magnifying our viewing scale until the curves look like straight lines.
In other words, if the curves are regular (have tangent lines) at $p$, it's reasonable to define the angle at which they meet to be the angle between their tangent lines.