MATHS PROJECT Quadrilaterals

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MATHS PROJECTQuadrilaterals

• Geometry EOC

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Definition

• A plane figure bounded by four line segments
  AB,BC,CD and DA is called a quadrilateral.

A
B
C
D
QuadrilateralI have exactly four sides.
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In geometry, a quadrilateral is a polygon with
four sides and four vertices. Sometimes, the
term quadrangle is used, for etymological
symmetry with triangle, and sometimes tetragon
for consistence with pentagon. There are over
9,000,000 quadrilaterals. Quadrilaterals are
either simple (not self-intersecting) or complex
(self-intersecting). Simple quadrilaterals are
either convex or concave.
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Types of Quadrilaterals

• Parallelogram
• Trapezium
• Kite

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Parallelogram
I have2 sets of parallel sides 2 sets of equal sidesopposite angles equal adjacent angles supplementarydiagonals bisect each otherdiagonals form 2 congruent triangles
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Types of Parallelograms
RectangleI have all of the properties of the parallelogram PLUS- 4 right angles- diagonals congruent
RhombusI have all of the properties of the parallelogram PLUS- 4 congruent sides- diagonals bisect angles- diagonals perpendicular
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SquareHey, look at me!I have all of the properties of the parallelogram AND the rectangle AND the rhombus.I have it all!
                                              
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Is a square a rectangle? Some people define
categories exclusively, so that a rectangle is a
quadrilateral with four right angles that is not
a square. This is appropriate for everyday use of
the words, as people typically use the less
specific word only when the more specific word
will not do. Generally a rectangle which isn't a
square is an oblong. But in mathematics, it is
important to define categories inclusively, so
that a square is a rectangle. Inclusive
categories make statements of theorems shorter,
by eliminating the need for tedious listing of
cases. For example, the visual proof that vector
addition is commutative is known as the
"parallelogram diagram". If categories were
exclusive it would have to be known as the
"parallelogram (or rectangle or rhombus or
square) diagram"!
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Trapezium I have only one set of parallel
sides. The median of a trapezium is parallel to
the bases and equal to one-half the sum of the
bases.
 
                                    
 
                                                                                                                                                                                                                                                                       
Trapezoid Regular Trapezoid
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Kite
It has two pairs of sides. Each pair is made up
of adjacent sides (the sides meet) that are equal
in length. The angles are equal where the pairs
meet. Diagonals (dashed lines) meet at a right
angle, and one of the diagonal bisects (cuts
equally in half) the other.
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Angle Sum Property Of Quadrilateral
The sum of all four angles of a quadrilateral is
360..
A
D
1
6
5
2
4
3
B
C
Given ABCD is a quadrilateral To Prove Angle
(ABCD) 360. Construction Join diagonal BD
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Proof In ABD Angle (126)180 -
(1) (angle sum property of ) In
BCD Similarly angle (345)180 (2) Adding (1)
and (2) Angle(126345)180180360 Thus,
Angle (ABCD) 360
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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.
A
3
1
E
D
F
2
4
C
B
Given In ABC. D and E are the mid-points of
AB and AC respectively and DE is joined To
prove DE is parallel to BC and DE1/2 BC
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• Construction Extend DE to F such that DeEF 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
• ADCF (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)
• ADCF (proved)
• Also ADDB (given)
• Thus, DBFC -(2)
• From (1) and(2)
• DBCF is a gm
• Thus, the other pair DF is parallel to BC and
  DFBC (By construction E is the mid-pt of DF)
• Thus, DE1/2 BC

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THE END