Example 1. $P$ is a point inside an equilateral $\triangle A B C$, and the ratios of the sizes of $\angle A P B, \angle B P C, \angle C P A$ are 5:6:7. Then, the ratio of the sizes of the three interior angles of the triangle with sides $P A, P B, P C$ is () (from smallest to largest).
(A) $2: 3: 4$
(B) $3: 4: 5$
(C) $4: 5: 6$
(D) cannot be determined
(1988, National Junior High School Mathematics Correspondence Competition, Selected Provinces and Cities)
From the given, we know that $\angle A P B=100^{\circ}, \angle B P C=$ $120^{\circ}, \angle C P A=140^{\circ}$.

From the proof of the previous problem, we know that the three interior angles of the triangle with sides of lengths $P A, P B, P C$ are (note the comparison and observation of the angles at one point):
$$
100^{\circ}-60^{\circ}, 120^{\circ}-60^{\circ}, 140^{\circ}-60^{\circ} \text {, }
$$

which are $40^{\circ}, 60^{\circ}, 80^{\circ}$, respectively. Thus, their ratio is $2: 3: 4$, so the answer is A.