Example 3 In $\triangle A B C$, $A B=A C, \angle A=$ $20^{\circ}, D$ is a point on $A C$, $\angle D B C=60^{\circ}, E$ is a point on $A B$, $\angle E C B=50^{\circ}$. Find the degree measure of $\angle B D E$.
Solution: As shown in Figure 3, let $E^{\prime}$ be the symmetric point of $E$ with respect to $AC$. Then $E^{\prime}A$, $E^{\prime}B$, $E^{\prime}C$, $E^{\prime}D$, and $E^{\prime}E$ are formed. It is known that $\triangle E^{\prime}EC$ is an equilateral triangle, $BE^{\prime}$ is the perpendicular bisector of $EC$, and $D$ is the incenter of $\triangle ABE^{\prime}$. Therefore,
$$
\begin{aligned}
\angle AE^{\prime}D & =\frac{1}{2} \angle AE^{\prime}B \\
& =50^{\circ} .
\end{aligned}
$$

It follows that $\angle AED=50^{\circ}$.
Thus, $\angle BDE=\angle AED$
$$
-\angle ABD=30^{\circ} \text{. }
$$