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Trapezoid Midsegment Theorem

Rule

Trapezoid Midsegment Theorem

The midsegment of a trapezoid is parallel to each base and its measure is one half the sum of the lengths of the bases.

If is the midsegment of trapezoid then

Proof

Proof

Let be a trapezoid and be its midsegment.

Draw the line containing the base and the line passing through and Let be the intersection point between these two lines.

The Vertical Angles Theorem gives that Also, since then because of the Alternate Interior Angles Theorem.

With the information written before and the Angle-Side-Angle Congruence Theorem, it can be concluded that The right-hand side relation above implies that is the midpoint of In consequence, is a midsegment of Thus, the Triangle Midsegment Theorem leads to the following conclusion. Because then is parallel to each base of the trapezoid. Using the Segment Addition Postulate can be rewritten as follows. Since then Finally, substituting the equation above into the equation given by the Triangle Midsegment Theorem, it will be obtained the desired result.