  Trapezoid Midsegment Theorem
Press on the numbers to see the steps of the proof.
1
Draw Trapezoid $$\mathtt{PQRS}$$, with $$\mathtt{\overline{XY}}$$ bisecting each of the non-parallel sides. Side $$\mathtt{\overline{PQ}}$$ is parallel to side $$\mathtt{\overline{SR}}$$.
2
Extend $$\mathtt{\overline{SR}}$$ to meet $$\mathtt{\overleftrightarrow{PY}}$$ at $$\mathtt{T}$$. Vertical angles are congruent, and alternate interior angles are congruent.
3
$$\mathtt{\Delta PQY \cong \Delta TRY}$$ by Angle-Side-Angle, so $$\mathtt{\color{green}{\overline{PY}} \cong \color{green}{\overline{TY}}}$$.
4
Thus, $$\mathtt{\color{purple}{\overline{XY}}}$$ is a midsegment of $$\mathtt{\color{brown}{\Delta PST}}$$, which means that $$\mathtt{\overline{XY} \parallel \overline{ST} \parallel \overline{SR}}$$ and $$\mathtt{XY = \frac{1}{2}ST}$$.
5
$$\mathtt{XY = \frac{1}{2}(SR + RT)}$$, and since $$\mathtt{RT = PQ}$$, $$\mathtt{XY = \frac{1}{2}(SR + PQ)}$$. And $$\mathtt{\overline{XY} \parallel \overline{PQ} \parallel \overline{SR}}$$.
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