Question

prove that the line joining the midpoints of non parallel sides of a trapezium is parallel to the parallel sides ​

Answer 1

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Without loss of generality, |AD|<|BC|. Draw a line passing through A and meeting BC at G and EF at H. Since AGCD is a parallelogram, H is the midpoint of AG. Note that triangles AEH and ABG are similar, hence, EH is parallel to BG, and the result follows.