# Trapezium Midsegment Theorem

A question from an ACT Math test:

Please teach me a way to get the answer quickly, I must've missed some properties of trapezoid cause I can only think of making similar triangles to find the answer but I'll have to assume two unknown numbers, which is too time-consuming.

$$ABCD$$ is a trapezoid with $$AB||CD$$. $$EF$$ is a line connecting the midpoints of sides $$AD$$ and $$BC$$, that is $$AE=ED$$ and $$BF=FC$$. Then, $$EF||DC$$ and that $$EF=\frac{1}{2}(AB+DC)$$.
Use this to go through the following solution: $$\overline{EF}=\frac{\overline{BC}+\overline{AD}}{2}\Rightarrow \overline{AD}=38\mathrm{~inches},\\\overline{GH}=\frac{\overline{EF}+\overline{AD}}{2}=31.5\mathrm{~inches}.$$