## 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

Here, AB∥CD and AD∥BC

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

- 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 _______.

### Concept Map

### 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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