Final answer:
A parallelogram with congruent diagonals is a rectangle because the congruent diagonals create congruent triangles, leading to all interior angles being right angles, fulfilling the definition of a rectangle.
Step-by-step explanation:
Demonstrating a Parallelogram is a Rectangle When Diagonals are Congruent
To prove that a parallelogram is a rectangle if its diagonals are congruent, we rely on the properties of a parallelogram and the definition of a rectangle.
In a parallelogram, opposite sides are equal in length, and opposite angles are equal. If the diagonals are congruent, each diagonal splits the parallelogram into two congruent triangles. By the Side-Side-Side criterion for triangle congruence, one can show that corresponding angles of these triangles are equal. Therefore, each pair of consecutive angles in the parallelogram would be supplementary and congruent, which is possible only when these angles are right angles. Thus, all the interior angles of the parallelogram are right angles, and by definition, the parallelogram is a rectangle.
The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side (hypotenuse). This is relevant here because if each pair of triangles formed by the diagonals is congruent and contains a right angle, then the four angles at which the diagonals intersect each other in the parallelogram are also right angles.