From Parallelogram to Square

To demonstrate that the parallelogram is a square, we must first prove that it is a rectangle or a rhombus.

Showing that at least one of the following conditions is met:

1. If the diagonals of the parallelogram are identical, it means that it is a rectangle.
2. If the parallelogram has an angle of 90 degrees, it means that it is a rectangle.

Showing that at least one of the following conditions is met:

1. If the parallelogram has a pair of equal adjacent sides, it is a rhombus.
2. If the diagonals of the parallelogram are bisectors, it is a rhombus.
3. If the diagonals of the parallelogram are perpendicular and form a 90º angle, it is a rhombus.

Observe, you should not prove that the parallelogram is both a rectangle and a rhombus.
Check which figures (rectangle or rhombus) you can prove exist in the following parallelogram, then proceed to the second step.

Now, we must demonstrate that the rectangle or rhombus before us is a square.
The second step is divided into two parts.

We can demonstrate that the rectangle is a square when at least one of the following conditions is met:

1. If there is a pair of adjacent sides that are equal, it is a square.
2. If the diagonals of the rectangle are perpendicular (form a 90-degree angle), it is a square.
3. If the diagonals of the rectangle are bisectors, it is a square.

We can prove that a rhombus is a square when at least one of the following conditions is met:

1. If the rhombus has a right angle (90º), it is a square.
2. If the diagonals of the rhombus are equal, it is a square.

Conclusion:
You will be able to prove that a parallelogram is a square if you first prove that it is a rectangle or a rhombus.
After you prove that the parallelogram is a rectangle or a rhombus, you will prove that said figure is a square.