1. MATHS PROJECT Quadrilaterals
Geometry EOC

2. *Quadrilateral I have exactly four sides.
Definition
A plane figure bounded by four line segments AB,BC,CD and DA is called a quadrilateral. A B C D
*Quadrilateral I have exactly four sides.

3. 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.

4. Types of Quadrilaterals
Parallelogram
Trapezium
Kite

6. Parallelogram I have: 2 sets of parallel sides
2 sets of equal sides opposite angles equal
adjacent angles supplementary diagonals bisect each other diagonals form 2 congruent triangles

7. Types of Parallelograms
*Rectangle I have all of the properties of the parallelogram PLUS - 4 right angles - diagonals congruent
*Rhombus I have all of the properties of the parallelogram PLUS - 4 congruent sides - diagonals bisect angles - diagonals perpendicular

8. *Square Hey, look at me! I have all of the properties of the parallelogram AND the rectangle AND the rhombus. I have it all!
                                              

9. 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"!

10. 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

11. 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.

12. 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 (A+B+C+D) =360.
Construction: Join diagonal BD

13. 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( )= =360
Thus, Angle (A+B+C+D)= 360

14. 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 B C
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 DE=1/2 BC

15. 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

16. THE END