Answer
1. The median of a trapezoid is a set distance away from the top and bottom of the trapezoid.
2. Since the distance between the median and the bases is constant, it must be parallel to the bases.
Work Step by Step
In a trapezoid ABCD where M is the midpoint of segment AB and N is the midpoint of segment DC.
1-extend the segment AD to AT, where D is between AT.
2- draw segment BT passing through N.
3- proving triangles BCN = triangle TDN
a- $\angle NDR = \angle NCB $ ( alternate interior angles )
b- CN= ND
c- $\angle BNC= \angle TND $ vertical angles.
Then the triangles are congruent by ASA.
4- by CPCTC BN=NT
So in triangle ABT, MN = 1/2 AT.
5- by theorem if the segment that joining two midpoints of the triangle segments then its equal half the base and parallel to the base. So MN is parallel to AT.
6-if AD// BC, MN//AD,then MN//BC. By transitive property
