Example 10 In $\triangle ABC$, there is an interior angle of $36^{\circ}$. A line $AD$ through vertex $A$ divides this triangle into two isosceles triangles. Then the number of different shapes of $\triangle ABC$ that satisfy the above conditions (similar triangles are considered the same shape) is ( ).
(A) 2
(B) 3
(C) 4
(D) 5
Solution: (1) When $\angle \mathrm{A}=$
$36^{\circ}$, let $\angle C=$
$\frac{1}{3} \angle A=12^{\circ}$. Then $\angle \mathrm{A}$
has a trisector that meets
the problem's requirements, as shown in Figure 6-1.
(2) When $\angle A \neq 36^{\circ}$, assume $\angle B=36^{\circ}$, then $\angle A D C=72^{\circ}, \angle C=72^{\circ}$ or $\angle C=54^{\circ}$; or $\angle A D C=108^{\circ}, \angle C=36^{\circ}$; or $\angle A D C=144^{\circ}$, $\angle C=18^{\circ}$. Respectively as shown in Figures 6.2-6-5.

In summary, the correct choice is (D).