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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