1. After cutting off a triangle from a convex $n$-sided cardboard, the remaining polygon has an interior angle sum of $2160^{\circ}$. Then the value of $n$ ( ).
(A) can only be 12
(B) can only be 13
(C) can only be 14
(D) None of the above is correct
-1 (D).
The number of sides of a polygon with an interior angle sum of $2160^{\circ}$ is
$$
2160^{\circ} \div 180^{\circ}+2=14 \text { (sides). }
$$
(1) As shown in Figure 2, when cutting $\triangle A B C$ through two vertices $A$ and $C$ of an $n$-sided polygon, $n=14-1+2=15$.
(2) As shown in Figure 2, when cutting $\triangle A B P$ through one vertex $A$ of an $n$-sided polygon,
$$
\begin{aligned}
n & =14-1+1 \\
& =14 .
\end{aligned}
$$
$n=14 \cdots 1=13$.