Which quadrilateral must have congruent Diagonals?
Both rectangles and isosceles trapezoids will always have congruent diagonals.
In the case of rectangles, this is the basis of the carpenters method of determining if a door or window frame is "square". If the diagonals are congruent, assuming the sides are congruent and the top/bottom of the frame is congruent, then the opening must be a rectangle.
A parallelogram with congruent diagonals must be a rectangle. Given parallelogram ABCD with AC=BD, then triangle ADC is congruent to triangle BCD, which implies that angle D is congruent to angle C. But this is a parallelogram, so angles B and C are supplementary, so they are right angles.