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Properties of Squares

Sep 10, 2022
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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. 

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. 

Theorem 

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

Given: AC=BD and AC⊥BD

To prove: ABCD is a square 

parallel

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,

parallel

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

Concept Map

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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