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 

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

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 _______.

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.