This is the fourth post of Geometry Concept series in which we will learn about the quadrilaterals and parallelograms: rectangle, rhombus and square and trapezium and their properties.

A four-sided closed figure is called a quadrilateral. ABCD is quadrilateral.

1. Area of a Quadrilateral = ½ X one of the diagonals X sum of perpendiculars drawn to the diagonals from the opposite vertices.

**Quadrilateral**A four-sided closed figure is called a quadrilateral. ABCD is quadrilateral.

1. Area of a Quadrilateral = ½ X one of the diagonals X sum of perpendiculars drawn to the diagonals from the opposite vertices.

So, Area of ABCD = ½ X AC X (DE + FB)

1. Sum of four interior angles is 360°. ie., ∠A + ∠B + ∠A + ∠D = 360°.

2. Sum of opposite sides of a quadrilateral circumscribed about a circle is always equal.

3. Figure formed by joining the mid-points of a quadrilateral is parallelogram.

A quadrilateral whose opposite side are parallel is known as a parallelogram.

1. Area of a Parallelogram

A parallelogram in which all the four angles are right angles is a rectangle.

1. Area of a Rectangle = Length X Breadth

Area of a Rhombus = Product of adjacent sides X sine of the included angle

2. Perimeter of a Quadrilateral = Sum of all sides

So, Perimeter of ABCD = AB + BC + CD + DA

**Properties of a Quadrilateral**

1. Sum of four interior angles is 360°. ie., ∠A + ∠B + ∠A + ∠D = 360°.

2. Sum of opposite sides of a quadrilateral circumscribed about a circle is always equal.

3. Figure formed by joining the mid-points of a quadrilateral is parallelogram.

**Parallelogram**

A quadrilateral whose opposite side are parallel is known as a parallelogram.

1. Area of a Parallelogram

**=**Base X Height or Product of any two sides X sine of the included angle

So, Area of ABCD = AB X DE.

2. Perimeter = Sum of all sides

So, Perimeter of ABCD = AB + BC + CD + DA

**Properties of a Parallelogram**

- Opposite sides are parallel and equal.
- Opposite angles are equal.
- Sum of two adjacent angles is 180°.
- Diagonals bisect each other.
- Each diagonal divided a parallelogram into two congruent triangles.
- Lines joining the mid-points of adjacent sides of a quadrilateral is a parallelogram.
- Lines joining the mid-points of adjacent sides of a parallelogram is a parallelogram.
- Parallelogram that is inscribed in a circle is a rectangle.
- Parallelogram that is circumscribed about a circle is rhombus.
- Parallelograms that lie on the same base and between the same parallel lines are equal in area.
- Area of a triangle is half the area of a parallelogram which lie on the same base and between the same parallel lines.
- A parallelogram is rectangle if its diagonals are equal.

**Rectangle**

A parallelogram in which all the four angles are right angles is a rectangle.

1. Area of a Rectangle = Length X Breadth

- Opposite sides are parallel and equal.
- Opposite angles are 90°.
- Diagonals are equal and bisect each other.
- When a rectangle is inscribed in a circle, the diameter of the circle is equal to the diagonal of the rectangle.
- For the given perimeter of rectangle, a square has maximum area.
- Figure formed by joining the mid-points of the adjacent sides of a rectangle is a rhombus.
- Quadrilateral formed by joining the mid-points of a of intersection of the angle bisectors of parallelogram is a rectangle.
- Every rectangle is parallelogram.

**Rhombus**

A parallelogram in which all the four sides equal.

1. Area of a Rhombus = ½ X product of the diagonals

or

1. Area of a Rhombus = ½ X product of the diagonals

or

Area of a Rhombus = Product of adjacent sides X sine of the included angle

**Properties of a Rhombus**

- Opposite sides are parallel and equal.
- Opposite angles are equal.
- Diagonals bisect each other at right angles.
- Diagonals also bisect vertex angles.
- Sum of any two adjacent angles is 180°.
- Figure formed by joining the mid-points of adjacent sides of a rhombus is rectangle.
- A parallelogram is rhombus if its diagonals are perpendicular to each other.

**Square**

A rectangle whose all sides are equal or a rhombus whose angles are right angles is called a square.

1. Area of a Square = (side)² = a² = (diagonal)² / 2

where diagonal = side√2

2. Perimeter = Sum of all sides

where diagonal = side√2

2. Perimeter = Sum of all sides

**Properties of a Square**- All sides are equal and of course opposite sides are parallel.
- All angles are right angles.
- Diagonals are equal and bisect each other at right angle.
- Diagonal of an inscribed square is equal to diameter of the inscribing circle.
- Side of a circumscribed square is equal to the diameter of the inscribed circle.
- The figure formed by joining the mid-points of the adjacent sides of a square is a square.

**Trapezium**

A quadrilateral whose only one pair of sides is parallel and other two sides are not parallel.

1. Area of a trapezium = ½ X (sum of parallel sides) X height

i.e., Area of ABCD = ½ X (AB + CD) X DM

2. Perimeter of trapezium = sum of all sides

i.e., Area of ABCD = ½ X (AB + CD) X DM

2. Perimeter of trapezium = sum of all sides

**Properties of Trapezium**- The line joining the mid-point of non-parallel sides is half the sum of the parallel sides and is called the median.
- If the non-parallel sides are equal then the diagonals will also be equal to each other.
- Diagonals intersect each other proportionally in the ratio of lengths of parallel sides.
- By joining the mid-points of adjacent sides of a trapezium four similar triangle are formed.
- If a trapezium is inscribed is a circle then its non-parallel sides are equal.
- AC² + BD² = BC² + AD² + 2 AB.CD

**Kite**

A kite is a figure having two pairs of adjacent sides are equal.

**Properties of Kite**

- AB = BC and AD = CD
- Diagonals intersect at right angles.
- Shorter diagonal is bisected longer diagonal.
- Area = ½ X product of diagonals

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