How many quadrilaterals are in a trapezium whose top corners drop upright lines to the base, with both diagonals drawn in the middle square?

With the whole top edge as a side: the square, the whole trapezium, the square plus one end triangle (2), and the trapeziums cut off by a diagonal (2), 6. Two more four-sided pieces each join an end triangle to the quarter of the square beside it. Total $6+2=8$.