Properties of Quadrilaterals

Properties of Quadrilaterals:

Angle Sum Property of a Quadrilateral: The sum of all the angles of a quadrilateral is 360°.

Parallelograms: A quadrilateral is called a parallelogram, if both the pairs of its opposite sides are parallel.

Properties of Parallelograms: 

• Opposite sides of a parallelogram are equal.
• Opposite angles of a parallelogram are equal.
• Diagonals of a parallelogram bisect each other.
• A diagonal of a parallelogram divides it into two congruent triangles.
• If the diagonal of a parallelogram bisects one of the angles of a parallelogram, it also bisects the second (opposite) angle.
• The angle bisectors of a parallelogram form a rectangle.

Conditions for a parallelogram to be a Quadrilateral:

• A quadrilateral is a parallelogram if it’s opposite sides are parallel.
• A quadrilateral is a parallelogram if it’s opposite angles are equal.
• If the diagonals of a parallelogram bisect each other, then the quadrilateral is a parallelogram.
• A quadrilateral is a parallelogram, if one of its sides are equal and parallel.

Summarization: 

A quadrilateral is parallelogram if:

• If both pairs of opposite sides are equal.
• If both pairs of opposite angles are equal.
• If diagonals bisect each other.
• If a pair of opposite sides are parallel as well as equal.

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Rectangle: A parallelogram in which each angle is a right angle i.e. 90° is called a rectangle.

Properties of Rectangle: 

• The diagonals of a rectangle are equal.
• Each of the four angles of a rectangle is 90°.
• Diagonals of a rectangle bisect each other.
• The angle bisectors of a parallelogram form a rectangle.

Square: A parallelogram in which all sides are equal and each angle is equal to 90° is called a square.

Properties of Square: 

• The diagonals of a square are equal.
• Diagonals of a square bisect each other at right angles.
• The quadrilateral formed by joining the midpoints of adjacent sides of a square is also a square.

Rhombus: A parallelogram in which all sides are equal is called as rhombus.

Properties of Rhombus: 

• The diagonals of a rhombus are perpendicular to each other.
• Diagonals of a rhombus bisect each other.
• The quadrilateral formed by joining the midpoints of adjacent sides of a rectangle is a rhombus.

Trapezium: A quadrilateral having exactly one pair of parallel sides is called a trapezium.

Isosceles Trapezium: A trapezium whose non-parallel sides are equal is called as isosceles trapezium.

Kite: A quadrilateral is called a kite if it has two pairs of equal adjacent sides but unequal opposite sides.

Some Important Points: 

• A square is rectangle and also a rhombus.
• A parallelogram is a trapezium.
• A kite is not a parallelogram.
• A rectangle or rhombus is not a square.
• A trapezium is not a parallelogram (as only one pair of opposite sides is parallel in a trapezium and we require both sides to be parallel in parallelogram).
• The sum of two adjacent angles of a parallelogram is 180°.

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