Notes
two pentagons solution

Two Pentagons

Solution by Transformation and Similar Triangles

With the points labelled as above, rotation about point $E$ of angle $72^\circ$ followed by a uniform scaling by the golden ratio also from $E$ takes $A$ to $B$ and $C$ to $D$. Therefore, applying this transformation to triangle $A E C$ produces triangle $B E D$ and so these triangles are similar. This means that $B D$ is obtained from $A C$ by rotation of $72^\circ$ (and scaling) and so angle $A \hat{F} B$ is $72^\circ$. The requested angle is therefore $108^\circ$.

Solution by Invariance Principle

The above diagram contains three special cases of the diagram. The yellow pentagon is common to all three. The pink and green each share an edge with the yellow, while the orange is such that at the common vertex then the sides of it and the yellow form straight lines. Using the angles in a regular pentagon, the required angle can be seen to be $108^\circ$.