Example 1. In isosceles trapezoid $\mathrm{ABCD}$, the lower base $\mathrm{AB}$ is 20, the upper base $\mathrm{CD}$ is 12, and the height is $8 \sqrt{2}$. By folding the trapezoid along the perpendicular bisector MN (Figure 1) of the bases to form a dihedral angle of $120^{\circ}$, find the length of $\mathrm{AC}$ at this time.
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Figure 2 is the folded shape. Figure 3 is the shape after Figure 2 is supported, where $\mathrm{AE}$ $\mathrm{FM}-\mathrm{PQCN}$ is a right quadrilateral prism, with the two bases being parallel parallelograms (the figure is intuitive, and the construction method is not described again, the same below).
\[
\begin{array}{c}
\mathrm{AM}=10, \\
\mathrm{MF}=\mathrm{NC}=6, \\
\angle \mathrm{AMF}=120^{\circ} \\
\mathrm{AF}^{2}-\mathrm{AM}^{2}+\mathrm{MF}^{2}-2 \mathrm{AM} \cdot \mathrm{MF} \cdot \cos 120^{\circ} \\
=14^{2} \cdot \mathrm{CF}=\mathrm{MN}=8 \sqrt{2}, \\
\therefore \quad \mathrm{AC}=\sqrt{\mathrm{AF}^{2}+\mathrm{CF}^{2}}=18 .
\end{array}
\]
Cosine Rule
If the angle between the skew lines $A M$ and $C N$ is $\theta$, and $\mathrm{MN}$ is the distance between the two skew lines, then
\[
\mathrm{AC}=\sqrt{\mathrm{MN}^{2}+\mathrm{AM}^{2}+\mathrm{CN}^{2}-2 \mathrm{AM} \cdot \mathrm{CN} \cos \theta}
\]

This result is consistent with the formula on page 46 of the stereometry textbook for key middle schools with a six-year system.
\[
\begin{array}{l}
M B=2 \sqrt{2}, \\
Q B=4, \\
\text { Pythagorean Theorem } \\
\mathrm{MB} / / \mathrm{PA} \perp \mathrm{QB} \\
Q M^{2}=24 \\
\mathrm{MR}=2, \\
Q R=2 \sqrt{3}, \\
\text { Chord Theorem } \\
\angle \mathrm{QRM}=\theta \\
Q M^{2}=16-8 \sqrt{3} \cos \theta. \\
\mathrm{PA}=4 \sqrt{2}, \\
\angle \mathrm{PLA} \text { is a right angle, } \because \mathrm{PL}=4, \\
\angle \mathrm{PAL}=45^{\circ} \\
\left.\begin{array}{l}
\mathrm{BR}=2, \\
\mathrm{AB}=6
\end{array}\right\} \Rightarrow \mathrm{LR}=4. \\
\therefore \quad \mathrm{PQ}=\sqrt{\mathrm{LR}^{2}+\mathrm{PL}^{2}+\mathrm{QR}^{2}-2 \mathrm{PL} \cdot \mathrm{QR} \cos \theta}=2 \sqrt{15}. \\
\end{array}
\]
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