How to draw tangents to a curve and obtain the corresponding angle?

I’m trying to solve an equation and plot the solutions in the complex plane:

   Solve[Rationalize[E^(-b (1.365)) + E^(-b (-0.350)) + E^(-b (-0.378)), 0] == 0, b]    ListPlot[ReIm[b /. % /. C[1] -> 0], ImageSize -> Large] 

Here is the plot:

enter image description here

Now, I need to draw tangent lines on the middle curve at the point where it is crossing the real axis, and obtain the angle. It is possible to do it by hand on the paper, but can one also do it by Mathematica?