Which quadrilateral must have congruent Diagonals?

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embizze | High School Teacher | (Level 2) Educator Emeritus

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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.