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No. Even in the non-US use of the term (a quadrilateral with at least one set of parallel lines), the lengths of the parallel lines may not be the same, and/or the angles formed by each adjacent side may be different (as in a rhomboid), resulting in diagonals of extremely different lengths.

Only in rectangles are diagonals "always" of equal length.

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Is every property of parallelogram also a property of all rectangles?

No as for example the diagonals of a rectangle are equal in length whereas they are not equal in length in a parallelogram


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