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​​​​​ABCD is a trapezium AB || DC point P and Q are the midpoint of set AD and  BC respectively. Then prove that PQ || AB and PQ=1/2 AB+DC
AB and CD are parallel lines.   We are given P and Q are mid point of AD and BC respectively.

Line AD and Line BC are segmented equally by line PQ which is drawn between parallel lines AB and CD.

Hence lines AB, CD and PQ are parallel to each other.

in ΔABD, PR is the line parallel to AB and passing through the midpoint P of side AD.

Hence PR = (1/2)AB ...................(1)

Similarly, in ΔBCD, QR is parallel to CD and passing through midpoint Q of  side BC.
Hence QR = (1/2)CD ..................(2)

Hence from eqns.(1) and (2),  PR+QR = PQ = (1/2)[ AB + CD ]
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