#### 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