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1. MATHS
PORJECT
WORK
QUADRILATERALS

2. MADE BY : -
Nihal Gour
SUBMITTED TO :-
Mr. D. K. Chourasia
IX B

3.  The word quadrilateral is made of the
words quad (meaning "four") and lateral (meaning
"of sides").
 This Gives us a simple definition about a
quadrilateral : A polygon (a closed figure made of
only line segments) with four side is called a
quadrilateral.
INTRODUCTION

4. 4 vertices
A.
B.
C
.
D.
• 4 sides
• 4 angles
QUADRILATERALS

5. QUADRILATERALS
A.
B.
C
.
D.
360o
The sum of ALLALL the angles of a
quadrilateral is 360o
C
.
A.
D.
B.

6. QUADRILATERALS
A.
B.
C
.
D.
360o
The sum of ALLALL the angles of a
quadrilateral is 360o

7. Angle Sum Property Of
Quadrilateral (In Detail)
The sum of all four angles of a quadrilateral is 360
.
.
A
B C
D
1
2
3
4
6
5
Given: ABCD is a quadrilateral
To Prove: Angle (A+B+C+D)
=360
.
Construction: Join diagonal BD

8. Proof: In ABD
Angle (1+2+6)=180 - (1)
(angle sum property of )
In BCD
Similarly angle (3+4+5)=180 – (2)
Adding (1) and (2)
Angle(1+2+6+3+4+5)=180+180=360
Thus, Angle (A+B+C+D)= 360

9. DifferEnt
types
of
QUADRILATERALS

10. The TRAPEZIUMThe TRAPEZIUM

11. • One pair of opposite sides are parallel
The TRAPEZIUMThe TRAPEZIUM

12. The PARALLELOGRAM

13. The PARALLELOGRAMThe PARALLELOGRAM
• Opposite sides are equal
• Opposite sides are parallel
• Opposite angles are equal
• Diagonals bisect each other

14. The RHOMBUS

15. The RHOMBUS
All sides are equal
• Opposite sides are parallel
• Opposite angles are equal
• Diagonals bisect each other at 90°

16. The RECTANGLE

17. The RECTANGLE
Opposite sides are equal
• Opposite sides are parallel
• All angles are right angles (90o
)
• Diagonals are equal and bisect one another

18. The SQUARE

19. The SQUARE
All sides are equal
• Opposite sides are parallel
• All angles are right angles (90o
)
• Diagonals are equal and bisect one
another at right angles

20. •Each pair adjacent sides (the sides meet) are equal
in length.
•The angles are equal where the pairs meet.
•Diagonals (dashed lines) meet at a right angle
•The longer diagonal bisects (cuts equally in half)
the shorter diagonal.
Kite
A
B
C
D AC
bisected
BD

21. Taxonomic Classification
The taxonomic classification of quadrilaterals is illustrated by the
following graph.

22. Quadrilaterals Flow Chart (Simpler)
General Quadrilateral
4 sides, 4 angles
Trapezoid
Only 1 pair of
parallel sides
Parallelogram
Opposite sides are
parallel and congruent
Rectangle
A parallelogram
with 4 right angles
Rhombus
A parallelogram with
4 congruent sides
Square
A rectangle with 4
congruent sides
Kite

23. Note that…….
 A square, rectangle and rhombus are all
parallelograms.
 A square is a rectangle and also a rhombus.
 A parallelogram is a trapezium.
 A kite is not a parallelogram.
 A trapezium is not a parallelogram (as only
one pair of opposite sides is parallel in a
trapezium and we require both pairs to be
parallel in a parallelogram).
 A rectangle or a rhombus is not a square.

24. Cyclic quadrilateral: the four
vertices lie on a circumscribed circle.
Tangential quadrilateral: the four
edges are tangential to an inscribed
circle. Another term for a tangential
polygon is inscriptible.
Bicentric quadrilateral: both cyclic
and tangential.
Some other types of
quadrilaterals

25. Some Properties
of Parallelogram,
Rectangle,
Rhombus and
Square

26. • A diagonal of a parallelogram divides it into two congruent triangles.
• In a parallelogram,
(i) opposite sides are equal
(ii) opposite angles are equal
(iii) diagonals bisect each other
• A quadrilateral is a parallelogram, if
(i) opposite sides are equal, or
(ii) opposite angles are equal, or
(iii) diagonals bisect each other, or
(iv) a pair of opposite sides is equal and parallel
• Diagonals of a rectangle bisect each other and are equal and vice-
versa.
• Diagonals of a rhombus bisect each other at right angles and vice-
versa.
• Diagonals of a square bisect each other at right angles and are
equal, and vice-versa.

27. The Mid-Point Theorem
The line segment joining the mid-points of two sides of a
triangle is parallel to the third side and is half of it.
Given: In ABCD and E are the mid-points of AB and AC respectively
and DE is joined
To prove: DE is parallel to BC and DE=1/2 BC
1
3
2
4
A
D E F
CB

28. Construction: Extend DE to F such that De=EF and join CF
Proof: In AED and CEF
Angle 1 = Angle 2 (vertically opp angles)
AE = EC (given)
DE = EF (by construction)
Thus, By SAS congruence condition AED = CEF
AD=CF (C.P.C.T)
And Angle 3 = Angle 4 (C.P.C.T)
But they are alternate Interior angles for lines AB and CF
Thus, AB parallel to CF or DB parallel to FC-(1)
AD=CF (proved)
Also, AD=DB (given)
Thus, DB=FC -(2)
From (1) and(2)
DBCF is a
gm
Thus, the other pair DF is parallel to BC and DF=BC (By
construction E is the mid-pt of DF)
Thus, DE=1/2 BC