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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