Example 1 As shown in Figure 1, find the size of $\angle 1+\angle 2+\cdots+\angle 7$.

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【Analysis】It is easy to notice that $\angle 2$, $\angle 3$, and $\angle 4$ are located in one quadrilateral, and $\angle 5$, $\angle 6$, and $\angle 7$ are located in another quadrilateral. This leads us to consider the sum of the interior angles of these two quadrilaterals. As shown in Figure 1, we add $\angle 8$ and $\angle 9$.
$$
\begin{array}{l}
\text { By } \angle 2+\angle 3+\angle 4+\angle 9=360^{\circ}, \\
\angle 5+\angle 6+\angle 7+\angle 8=360^{\circ},
\end{array}
$$

Adding the two equations, we get
$$
\begin{array}{l} 
\angle 2+\angle 3+\cdots+\angle 9=720^{\circ} . \\
\text { Therefore, } \angle 1+\angle 2+\cdots+\angle 7 \\
=720^{\circ}+\angle 1-(\angle 8+\angle 9) \\
= 360^{\circ}+\left[\angle 1+\left(180^{\circ}-\angle 8\right)+\left(180^{\circ}-\angle 9\right)\right] \\
= 360^{\circ}+180^{\circ}=540^{\circ} .
\end{array}
$$