1. Quadrilateral
A quadrilateral is a four-sided figure.
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It has \(4\) sides.
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It has \(4\) vertices.
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It has \(4\) angles.
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The angles of a quadrilateral are the angles between its sides.
2. Rectangle
Definition
A rectangle is a quadrilateral in which all the angles are \(90°\).
Properties of a Rectangle
Remember these four important properties:
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All angles are \(90°\).
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Opposite sides are equal.
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Opposite sides are parallel.
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Diagonals are equal in length and bisect each other.
So, if the diagonals of a rectangle intersect at \(O\):
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\(AC = BD\)
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\(AO = OC\)
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\(BO = OD\)
Important Rectangle Idea
A rectangle can also be described as a quadrilateral whose:
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diagonals are equal, and
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diagonals bisect each other.
Thus:
Equal diagonals + diagonals bisect each other → Rectangle
Also, if all four angles of a quadrilateral are \(90°\), its opposite sides are automatically equal. Therefore, it is a rectangle.
Construction Idea to Remember
To form a rectangle using its diagonals:
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Draw two diagonals of equal length.
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Make them bisect each other.
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The angle between the diagonals can be different; it need not be \(90°\).
The resulting quadrilateral is a rectangle.
3. Square
Definition
A square is a quadrilateral in which:
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all four angles are \(90°\), and
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all four sides are equal.
Square as a Special Rectangle
A square is a special type of rectangle.
Therefore:
Every square is a rectangle, but every rectangle is not a square.
Properties of a Square
Remember all five:
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All four sides are equal.
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Opposite sides are parallel.
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All four angles are 90°.
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Diagonals are equal and bisect each other at \(90°\).
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Diagonals bisect the angles of the square.
Therefore, each angle of a square is divided into: \(45° + 45°\)
Diagonal Conditions for a Square
A quadrilateral will form a square when its diagonals:
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are equal in length,
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bisect each other, and
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intersect at right angles (\(90°\)).
So remember:
Equal diagonals + bisect each other + perpendicular → Square
4. Rectangle vs Square – Quick Comparison
Remember
Square = Rectangle + all sides equal
But:
Rectangle ≠ necessarily a square
5. Angles in a Quadrilateral
Angle Sum Property
The sum of the four interior angles of any quadrilateral is \(360°\).
Why?
A diagonal divides a quadrilateral into two triangles.
Each triangle has an angle sum of \(180°\).
Therefore:
\(180° + 180° = 360°\)
Hence,
Sum of angles of a quadrilateral \(= 360°\)
Finding a Missing Angle
If three angles of a quadrilateral are known:
Missing angle \(= 360° −\) sum of the three known angles
Important Result
A quadrilateral cannot have three right angles and a fourth angle that is not \(90°\).
If three angles are \(90°\):
\(90° + 90° + 90° = 270°\)
Therefore, the fourth angle must be:
\(360° - 270° = 90°\)
So, a quadrilateral with three right angles automatically has four right angles and is therefore a rectangle.