Example 4 Two isosceles triangles have supplementary vertex angles, one triangle has side lengths of $a, a, b (a > b)$, and the other triangle has side lengths of $b, b, a$. Find the degree measures of their interior angles.

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Explanation: Two isosceles triangles are combined to form the shape shown in Figure 3, where,
$$
\begin{array}{l}
A B= \\
A C=C D=a, B C= \\
B D=b .
\end{array}
$$

Since $\angle B A C + \angle D B C = 180^{\circ}$, the base angles must be complementary, i.e.,
$$
\angle A C B + \angle B C D = 90^{\circ}.
$$

Draw $B E \perp D C$ at point $E$, and $B F \perp A C$ at point $F$. Since $B F = E C = \frac{1}{2} D C = \frac{1}{2} a = \frac{1}{2} A B$, we have $\angle B A F = 30^{\circ}$, i.e., $\angle B A C = 30^{\circ}$. At this point, $\angle A B C = \angle A C B = 75^{\circ}$, $\angle B D C = \angle B C D = 15^{\circ}$, $\angle D B C = 150^{\circ}$.