Question

Prove if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Answer 1

Using the properties of a parallelogram and the properties of congruent triangles, the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Step-by-step explanation:

To prove that the diagonals of a parallelogram are congruent, the proof involves:

1. Start with a parallelogram ABCD.
2. Draw diagonals AC and BD, which intersect at point E.
3. Using the properties of a parallelogram, we know that opposite sides are congruent and parallel. Therefore, AB is congruent to DC and AD is congruent to BC.
4. Using the properties of congruent triangles, we can prove that triangle ABE is congruent to triangle CDE and triangle ADE is congruent to triangle CBE.
5. When two triangles are congruent, their corresponding sides are congruent.
6. Therefore, AE is congruent to CE and BE is congruent to DE.
7. Since the diagonals AC and BD intersect at point E and their corresponding sides are congruent, the diagonals of the parallelogram are congruent.

Therefore, if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.