$\overline{AD}=\overline{DE}\Rightarrow \hat{A}={{\hat{E}}_{1}}$ (1)
$\begin{align}
& \left. \begin{matrix}
D:\overline{AD}=\overline{DC} \\
\overline{AD}=\overline{DE} \\
\end{matrix} \right\}\Rightarrow \overline{DE}=\overline{DC} \\
& \Rightarrow \hat{C}={{{\hat{E}}}_{2}} \\
\end{align}$
$\begin{align}
& A\overset{\Delta }{\mathop{C}}\,E:{{\underbrace{\hat{A}+\hat{C}+{{{\hat{E}}}_{1}}+{{{\hat{E}}}_{2}}=180{}^\circ }_{{}}}_{{}}}\xrightarrow{(1),(2)}2{{{\hat{E}}}_{1}}+2{{{\hat{E}}}_{2}}=180 \\
& \Rightarrow {{{\hat{E}}}_{1}}+{{{\hat{E}}}_{2}}=90{}^\circ \Rightarrow A\hat{E}C=90{}^\circ \\
\end{align}$