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# Properties of Squares

## Key Concepts

• Define a square.
• Explain the properties of a square.
• Solve problems based on the properties.

## Square

A parallelogram with all equal sides and all angles equal to 90 degrees is a square.

### Properties of square

• The diagonals of a square are perpendicular and congruent.
• A diagonal bisects the opposite angles.

#### Theorem

If the diagonals of a parallelogram are perpendicular and congruent, then it is a square.

Let the sides of the figure be AB, BC, CD and AD.

Given: AC=BD and AC⊥BD

To prove: ABCD is a square

Since the opposite sides of the figure are parallel, so, it is in the shape of a parallelogram.

A parallelogram in which the diagonals are congruent is called a rectangle.

So, ABCD is a rectangle.         …(1)

Therefore,

∠A=∠B=∠C=∠D=90°

A parallelogram in which the diagonals are perpendicular is called a rhombus.

So, ABCD is a rhombus.         …(2)

Therefore, AB=BC=CD=DA

From (1) and (2),

A rectangle whose all sides are equal is a square.

Hence proved.

## Exercise

1. What is the value of p if STUV is a square?
• What is the value of x if ABCD is a square?
• Find the value of p if EFGH is a square.
• The perimeter of square DRUM is ________.
• If ASDF is a square, the value of k will be _______.

### What we have learned

• Square: A parallelogram with all equal sides and all angles equal to 90 degrees.
• The diagonals of a square are congruent and perpendicular.

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