In parallelogram ABCD, AC equals BD. Is ABCD a rectangle?

In a parallelogram ABCD, if the diagonals AC and BD are equal, it does not necessarily mean that ABCD is a rectangle. It does indicate, though, that ABCD is a special type of parallelogram — a rectangle or a rhombus.

For ABCD to be a rectangle, it must have four right angles (each being 90 degrees). Just knowing that the diagonals AC and BD are equal is not enough to ensure that all angles are right angles. It is a property of rectangles that their diagonals are equal in length, but having equal diagonals alone does not guarantee that all interior angles are right angles.

However, if a parallelogram has equal diagonals and is also known to have one right angle (or, due to the properties of a parallelogram, if one angle is right, then all angles are right), then it can be concluded that it is a rectangle. Without that additional piece of information about the angles, you cannot confirm that ABCD is a rectangle just from the fact that AC equals BD.