Example 6 As shown in Figure 5, in $\triangle A B C$, $\angle B=\angle C$, point $D$ is on side $B C$, $\angle B A D=50^{\circ}$. Take point $E$ on $A C$ such that $\angle A D E=$ $\angle A E D$. Then the degree measure of $\angle E D C$ is ( ).
(A) $15^{\circ}$
(B) $25^{\circ}$
(C) $30^{\circ}$
(D) $50^{\circ}$
Solution 1 Notice that
$$
\begin{array}{l}
\angle E D C=\angle A D C-\angle A D E \\
=(\angle B A D+\angle B)-\angle A E D \\
=\angle B A D+\angle B-(\angle E D C+\angle C) \\
=\angle B A D-\angle E D C .
\end{array}
$$

Therefore, $\angle E D C=\angle B A D \div 2=25^{\circ}$.
Solution 2 Let $\angle D A C=50^{\circ}$. Then
$$
\angle A D E=65^{\circ}, \angle A D C=90^{\circ} \text {. }
$$

Thus, $\angle E D C=25^{\circ}$.