Question

which statement can be used to prove that a given parallelogram is a rectangle?
all the sides of the parallelogram are equal in measure.
the consecutive angles of the parallelogram are supplementary.
the diagonals of the parallelograms are congruent.
the diagonals of the parallelogram are perpendicular.

Explanation:

• Analyze Option 1: A parallelogram with all sides equal is a rhombus, not necessarily a rectangle. So this is incorrect.
• Analyze Option 2: In any parallelogram, consecutive angles are supplementary (since opposite sides are parallel, consecutive angles are same - side interior angles and sum to \(180^{\circ}\)). This is a property of all parallelograms, not a distinguishing property of rectangles. So this is incorrect.
• Analyze Option 3: A theorem in geometry states that if the diagonals of a parallelogram are congruent (equal in length), then the parallelogram is a rectangle. This is a valid criterion for proving a parallelogram is a rectangle.
• Analyze Option 4: A parallelogram with perpendicular diagonals is a rhombus (or a square, which is a special case of a rhombus and a rectangle), but not all such parallelograms are rectangles. So this is incorrect.

Answer:

C. The diagonals of the parallelograms are congruent.