Example 5 As shown in Figure 1.4.6, in quadrilateral $A B C D$, $\angle B=\angle D=90^{\circ}$, $\angle A=60^{\circ}, A B=4, A D=5$, find $B C: C D$.
Extend $A D, B C$ to intersect at point $E$. Since $\angle B=\angle D=90^{\circ}$, we get two right triangles $\triangle A B E$ and $\triangle C D E$. According to the given conditions, $A E=2 A B=8$, $D E=3$.

In the right triangle $\triangle C D E$, we can find $C D=\sqrt{3}, C E=2 \sqrt{3}$. In the right triangle $\triangle A B E$, we can find $B E=4 \sqrt{3}, B C=2 \sqrt{3}$. Therefore,
$$
B C: C D=2 \sqrt{3}: \sqrt{3}=2
$$