Parallelograms Converse of Theorem 4

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Now consider the converse of this theorem.

Statement

If two diagonals of a parallelogram are equal, it is a rectangle.

Given:

ABCD is a parallelogram in which AC=BD.

To prove:

Parallelogram ABCD is a rectangle.

Proof:

In triangles ABC and DBC,

AB=DC (opposite sides of parallelogram)

BC=BC (common side)

AC=BD (given)

(corresponding parts of

corresponding triangles)

But these angles are consecutive interior angles on the same side of transversal BC and AB||DC.

By definition of rectangle, parallelogram ABCD is a rectangle.

Hence the theorem is proved.