Class 9: Quadrilaterals – Lecture Notes

$\displaystyle \textbf{Points to Remember}$

$\displaystyle \textbf{1. Quadrilateral}$
$\displaystyle \text{A closed four-sided figure is called a quadrilateral.}$
$\displaystyle \text{(i) It has four sides, four vertices, four angles and} \\ \text{two diagonals.}$
$\displaystyle \text{(ii) Sum of its four angles }=360^\circ,$
$\displaystyle \angle A+\angle B+\angle C+\angle D=360^\circ.$
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$\displaystyle \textbf{2. Types of Quadrilaterals}$
$\displaystyle \textbf{(1) Parallelogram}$
$\displaystyle \text{A quadrilateral in which opposite sides are parallel is} \\ \text{called a parallelogram.}$
$\displaystyle \text{If }AB\parallel DC\text{ and }AD\parallel BC, \\ \text{ then }ABCD\text{ is a parallelogram.}$
$\displaystyle \text{Its opposite sides are equal, i.e., }AB=CD \\ \text{ and }AD=BC.$
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$\displaystyle \textbf{(2) Rhombus}$
$\displaystyle \text{A parallelogram having all sides equal is called a} \\ \text{rhombus.}$
$\displaystyle \text{If }AB=BC=CD=DA,\text{ then }ABCD\text{ is a} \\ \text{rhombus.}$
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$\displaystyle \textbf{(3) Rectangle}$
$\displaystyle \text{A parallelogram each of whose angles measures } \\ 90^\circ\text{ is called a rectangle.}$
$\displaystyle \text{Its opposite sides are equal.}$
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$\displaystyle \textbf{(4) Square}$
$\displaystyle \text{A rectangle having all sides equal is called a square.}$
$\displaystyle \text{If }PQ=QR=RS=SP,\text{ then }PQRS\text{ is a} \\ \text{square.}$
$\displaystyle \text{Each of its angles is }90^\circ.$
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$\displaystyle \textbf{(5) Trapezium}$
$\displaystyle \text{A quadrilateral having one pair of opposite sides} \\ \text{parallel and the other pair}$
$\displaystyle \text{non-parallel is called a trapezium.}$
$\displaystyle \text{If }AB\parallel DC\text{ and }AD\not\parallel BC, \\ \text{ then }ABCD\text{ is a trapezium.}$
$\displaystyle \text{If the non-parallel sides are equal, it is called an} \\ \text{isosceles trapezium.}$
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$\displaystyle \textbf{(6) Kite}$
$\displaystyle \text{A quadrilateral in which two pairs} \\ \text{of adjacent sides are equal is called a kite.}$
$\displaystyle \text{If }AB=AD\text{ and }CB=CD,\text{ then } \\ ABCD\text{ is a kite.}$
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$\displaystyle \textbf{Results on Parallelogram}$
$\displaystyle \textbf{Theorem 1: }\text{In a parallelogram:}$
$\displaystyle \text{(i) Opposite sides are equal.}$
$\displaystyle \text{(ii) Opposite angles are equal.}$
$\displaystyle \text{(iii) Each diagonal bisects the} \\ \text{parallelogram.}$
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$\displaystyle \textbf{Theorem 2:}$
$\displaystyle \text{The diagonals of a parallelogram bisect each other.}$
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$\displaystyle \textbf{Theorem 3:}$
$\displaystyle \text{If one pair of opposite sides of a quadrilateral are} \\ \text{parallel and equal,}$
$\displaystyle \text{then the quadrilateral is a parallelogram.}$
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