Quadrilaterals

Synopsis

• Introduction to quadrilateral.
  A quadrilateral is a closed figure obtained by joining four points (with no three points collinear) in an order.
  
  Here, ABCD is a quadrilateral.

Parts of a quadrilateral

• A quadrilateral has four sides, four angles and four vertices.
• Two sides of a quadrilateral having no common end point are called its opposite sides.
• Two sides of a quadrilateral having a common end point are called its adjacent sides.
• Two angles of a quadrilateral having common arm are called its adjacent angles.
• Two angles of a quadrilateral not having a common arm are called its opposite angles.
• A diagonal is a line segment obtained on joining the opposite vertices.

• Angle sum property of a quadrilateral
  Sum of all the angles of a quadrilateral is 360º. This is known as the angle sum property of a quadrilateral.

• Types of quadrilaterals and their properties:
  
  
  

Important facts about quadrilaterals

• If the non-parallel sides of trapezium are equal, it is known as isosceles trapezium.
• Square, rectangle and rhombus are all parallelograms.
• Kite and trapezium are not parallelograms.
• A square is a rectangle.
• A square is a rhombus.
• A parallelogram is a trapezium.

Formation of a new quadrilateral using the given data

• If the diagonals of a parallelogram are equal, then it is a rectangle.
• If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
• If the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

If there are three or more parallel lines and the intercepts made by them on a transversal are equal, then the corresponding intercepts on any other transversal are also equal.

• Area of Polygons
• Area of a Quadrilateral
  Area of a quadrilateral  × One of the diagonals × Sum of perpendiculars drawn on the diagonals from opposite vertices.

Area of a Parallelogram

• Area of a parallelogram = base × height
• Area of a parallelogram = Product of any two adjacent sides × sine of included angle
• Perimeter of a parallelogram = 2 × Sum of any two adjacent sides
• Parallelograms that lie on the same base and between the same parallel lines are equal in area.

Area of a rectangle

• Area of a rectangle = length × breadth = l × b
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Area of a Rhombus

• Area of Rhombus = base × height
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• Area of Rhombus = Product of adjacent sides × sine of included angle 

Area of a Square

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• Perimeter of a Square =4× side

Area of a Trapezium

Area of a Kite

Cyclic Quadrilateral

• A quadrilateral whose all four vertices lie on a circle is called cyclic quadrilateral.
• The sum of pair of opposite angles of a cyclic quadrilateral is 180o.
• If a side of a cyclic quadrilateral is produced, then the exterior angle so formed is equal to the interior opposite angle.
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