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The following relation holds true for midsegment FE.
FE∥AB∥DC and FE=21(AB+CD)
Let ABCD be a trapezoid and FE be its midsegment. By definition, the midsegment of a trapezoid connects the midpoints of the nonparallel sides.
Next, draw the line connecting the points C and F, then extend the base AB such that it intersects with CF. Let G be the point of intersection between these two lines.
By the Vertical Angles Theorem, it is given that ∠AFG and ∠CFD are congruent angles. Also, since GB and CD are parallel, then ∠GAF and ∠D are congruent angles by the Alternate Interior Angles Theorem.
Triangles GAF and CDF have two pairs of congruent angles and a pair of included congruent sides. Therefore, by the ASA Congruence Theorem, it is concluded that △GAF≅△CDF. Since corresponding parts of congruent figures are congruent, then GA≅CD and FG≅FC.