Example 1: Through the midpoints $E$ and $F$ of the edges $AB$ and $BC$ of the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$, a section is made such that the angle between the section and the base is $45^{\circ}$. The shape of this section is ( ).
(A) Triangle or pentagon
(B) Triangle or hexagon
(C) Hexagon
(D) Triangle or quadrilateral
Explanation: As shown in Figure 1, it is clear that a section passing through points $E$ and $F$ must intersect the edge $B B_{1}$, and this section is a triangle.

Let the angle between the section passing through point $D_{1}$ and the base be $\alpha$. It is easy to find that
$$
\begin{array}{l}
\tan \alpha=\tan \angle D_{1} G D \\
=\frac{2 \sqrt{2}}{3}1 . \\
\text { Therefore, } \beta>45^{\circ} .
\end{array}
$$

Thus, the other section sought should intersect $A_{1} D_{1}$ and $D_{1} C_{1}$ (but not at their endpoints), forming a hexagon. Hence, the correct choice is (B).