1. If all the diagonals of a convex $n$-sided polygon $F (n \geqslant 4)$ are equal, then
A. $F \in\{$ quadrilateral $\}$
B. $F \in\{$ pentagon $\}$
C. $F \in\{$ quadrilateral $\} \bigcup\{$ pentagon $\}$
D. $F \in\{$ equilateral polygon $\} \bigcup\{$ equiangular polygon $\}$
Obviously, all diagonals of a rectangle and a regular pentagon are equal. When $n=6$, let's assume $A D=A C=B D \Rightarrow A B C D$ is an isosceles trapezoid. Similarly, $A D E F$ is also an isosceles trapezoid $\Rightarrow C E \perp A D \Rightarrow \alpha=30^{\circ}$. Thus, $\triangle C E F$ is a right triangle $\Rightarrow C F>C E$, which is a contradiction. Combining with the options, the answer is $C$.