We are going to transform $ABCDE$ to a regular shape using rotation. This can be done by rotating $\triangle{AED}$ around point $A$ clockwise till point $E$ moves to point $B$. This is possible because $AB=AE$.

Because $\angle{ABC} + \angle{ABD'} = \angle{ABC}+\angle{AED} = 180^\circ$, we conclude points $D'$, $B$, and $C$ are collinear. Now because $CD = BC+DE= BC+BD'=CD'$ and $AD'=AD$, we find $AD'CD$ is a kite which means $\angle{ADC}=\angle{AD'C}=\angle{ADE}$.