In the diagram below, each side of convex quadrilateral $ABCD$ is trisected. (For example, $AP = PQ = QB.$) The area of convex quadrilateral $ABCD$ is $180.$ Find the area of the shaded region.
Area of quadrilateral = AD *DC = 180
Shaded area (a trapezoid) = (1/2) AD * ( DC + DC/3) = (1/2) AD *(4/3)DC = (2/3) AD * DC
So....the shaded area = (2/3) (180) = 120
+412 
