Elementary Geometry for College Students (6th Edition)

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