Example 6 As shown in Figure 1.4.7, in trapezoid $A B C D$, $A B / / C D, \angle A+\angle B=90^{\circ}, A B=a, C D=$ $b, E, F$ are the midpoints of $A B, C D$ respectively, find the length of $E F$.
Given $\angle A+\angle B=90^{\circ}$, the trapezoid can be completed into a right triangle. Extend $A D, B C$ to intersect at $G$, then $\triangle A B G$ is a right triangle. Connect $G F$ and extend it to intersect $A B$ at $E^{\prime}$.

Since $A B / / C D$, we have $D F: A E^{\prime}=F C: E^{\prime} B$, and since $D F=$ $F C$, it follows that $A E^{\prime}=E^{\prime} B$, meaning $E^{\prime}$ coincides with $E$. Also,
$$
G E=\frac{1}{2} A B=\frac{1}{2} a, G F=\frac{1}{2} C D=\frac{1}{2} b
$$

Thus,
$$
E F=\frac{1}{2}(a-b)
$$