2. If all the diagonals of a convex $n(n \geqslant 4)$-sided polygon $F$ are equal, then $(\quad)$.
A. $F \in\{$ quadrilateral $\}$
B. $F \in\{$ pentagon $\}$
C. $F \in\{$ quadrilateral $\} \cup\{$ pentagon $\}$
D. $F \in\{$ equilateral polygon $\} \cup\{$ equiangular polygon $\}$
2. C.

If $n \geqslant 6$, construct a convex polygon $A_{1} A_{2} A_{3} \cdots A_{n}$, such that $A_{1} A_{3}=A_{1} A_{4}=A_{3} A_{n}=A_{4} A_{n}$, then $A_{1} A_{4}+A_{3} A_{n}=A_{1} A_{3}+A_{4} A_{n}$. By the triangle inequality, $A_{1} A_{4}+A_{3} A_{n}>A_{1} A_{3}+A_{4} A_{n}$, which is a contradiction. Therefore, $n<6$. Counterexamples: A regular pentagon can negate $A$, a square can negate $B$, and an isosceles trapezoid can negate $D$.