Example 10 In $\triangle A B C$, $\angle A B C=50^{\circ}$, $\angle A C B=30^{\circ}, R$ is a point inside the triangle, $\angle R A C=$ $\angle R C B=20^{\circ}$. Find the degree measure of $\angle R B C$.
Solution: As shown in Figure 10, let point $E$ be the reflection of point $R$ over $AC$, and point $D$ be the reflection of point $A$ over $EC$. Connect $DA$, $DR$, $DE$, $DC$, $EA$, and $EC$.
It is easy to see that $\triangle EDA$ is an equilateral triangle, so
$$
AD = AE = AR \text{.}
$$

In $\triangle ACD$, it is easy to see that $\angle DAC = 80^{\circ}$, which implies $\angle BAC + \angle DAC = 180^{\circ}$.
Thus, points $B$, $A$, and $D$ are collinear.
We have $\angle DCB = 50^{\circ} = \angle DBC$,
and $\angle BDC = 80^{\circ}$.
In $\triangle ARD$, given $\angle RAD = 100^{\circ}$, we know
$$
\angle RDA = \angle DRA = 40^{\circ} = \frac{1}{2} \angle BDC \text{.}
$$

From (1) and (2), we can conclude that points $B$ and $C$ are symmetric with respect to $DR$.
Therefore, $\angle RBC = \angle RCB = 20^{\circ}$.