PQRS is a trapezoid with QR || PS.
It is known that:
QR = 1000, PS = 2008, and:
∠SPQ = 37o, ∠PSR = 53o
Points M and N are the respective midpoints of sides QR and PS
Find the length of MN.
Extend PQ and RS to meet at T. Angle QTR (which is the same as angle PTS) is a right angle. Then triangles QTR and PTS are similar triangles.
T also lies on the line containing MN. That line is the median of angle QTR/PTS. The median from a right angle is congruent to half of the hypotenuse. Then TM = QR/2 = 500 and TN = PS/2 = 1004. Then MN = TN - TM = 504.