1. If all diagonals of a convex $n$-sided polygon $F(n \geqslant 4)$ are equal, then
(A) $F \in \mid$ quadrilateral $\mid$
(B) $F \in \mid$ pentagon $\mid$
(C) $F \in\{$ quadrilateral $\} \cup\{$ pentagon $\}$
(D) $F \in\{$ equilateral polygons $\} \cup\{$ equiangular polygons $\}$
Answer ( )
(C)
I. Multiple Choice Question
1.【Analysis and Solution】Using a regular pentagon as a counterexample can negate conclusion (A); using a square as a counterexample can negate conclusion (B); using an isosceles trapezoid as a counterexample can negate conclusion (D); therefore, the correct conclusion can only be (C). In fact, to affirm $(\mathrm{C})$ positively, it still needs to be proven that $nA_{1} A_{3} + A_{4} A_{n}$, so the number of sides of a convex polygon with all diagonals equal cannot be greater than or equal to 6, i.e., it can only be a quadrilateral or a pentagon. Examples of a square and a regular pentagon show that convex quadrilaterals and pentagons with all diagonals equal do exist.