Example 3 Let $P, A, B, C$ and $D$ be five distinct points in space such that $\angle A P B=\angle B P C=\angle C P D=\angle D P A=\theta$, where $\theta$ is a given acute angle. Determine the maximum and minimum values of $\angle A P C+\angle B P D$.
(13th US Olympiad Problem)
Solve As shown in Figure $24-4, \angle A P B+\angle B P C+\angle C P D+\angle D P A=4 \theta$. Let $\angle A P C=2 \theta_{1}, \angle B P D=2 \theta_{2}, P O$ be the projection of $P B$ on the plane $A P C$. By Corollary 4 of Property 3, we know $\angle A P O=\angle O P C=\theta_{1}$, and the plane $O P B \perp$ plane $A P C$. Therefore, $\angle D P O=\angle O P B=\theta_{2}$. Also, by Corollary 1 of Property 3, we know $\cos \theta_{1} \cdot \cos \theta_{2}=\cos \theta$, and $0<\theta_{1}, \theta_{2}<\theta$, thus $\angle A P C+\angle B P D>2 \theta$. From (*) we know this value can be arbitrarily close to $2 \theta$, when $\theta=0$, $\angle A P C+\angle B P D$ has a minimum value of 0.