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Rule

Trapezoid Midsegment Theorem

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

The following relation holds true for midsegment

and

Proof

Let be a trapezoid and be its midsegment. By definition, the midsegment of a trapezoid connects the midpoints of the nonparallel sides.

Next, draw the line connecting the points and then extend the base such that it intersects with Let be the point of intersection between these two lines.

By the Vertical Angles Theorem, it is given that and are congruent angles. Also, since and are parallel, then and are congruent angles by the Alternate Interior Angles Theorem.

Triangles and have two pairs of congruent angles and a pair of included congruent sides. Therefore, by the ASA Congruence Theorem, it is concluded that Since corresponding parts of congruent figures are congruent, then and
Since and are congruent, it can be stated that is the midpoint of and it is given that is the midpoint of Consequently, is a midsegment of Therefore, by the Triangle Midsegment Theorem, and are parallel and is half
Using the fact that and are parallel, the first part of the theorem can be obtained.
To obtain the second part, rewrite using the Segment Addition Postulate.
Since then
Finally, substituting the equation above into the equation given by the Triangle Midsegment Theorem, the desired result will be obtained.
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