Draw the pentagon, equilateral triangle ABG and line segment BF first, then draw CB equal in length to DF with angle ABC equal to 96$^\circ$ and C to the left of B. Add the label E as shown. Now draw CD. The symmetry of the pentagon about the line segment BF implies the angles shown at B, F and E. As EB = EF (they are opposite equal angles of triangle BEF), CE = DE, and so angles ECD and EDC each equal 54$^\circ$.

The above proves that the diagram is possible, so the initial guess that angle ADF is 108$^\circ$ is correct.