Quadrilaterals
Quadrilateral just means "four sides" (quad means four, lateral means side). Any foursided shape is a Quadrilateral. But the sides have to be straight, and it has to be 2dimensional. 
Properties
 Four sides (edges)
 Four vertices (corners)
 The interior angles add up to 360 degrees:
Try drawing a quadrilateral, and measure the angles. They should add to 360°
Types of Quadrilaterals
There are special types of quadrilateral:
Let us look at each type in turn:
The Rectangle
means "right angle"  
and
 show equal sides  
Also opposite sides are parallel and of equal length.
The Rhombus
Also opposite sides are parallel and opposite angles are equal.
Another interesting thing is that the diagonals (dashed lines in second figure) meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles.
The Square
means "right angle"  
show equal sides  
Also opposite sides are parallel.
A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).
The Parallelogram
NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!
Example:
A parallelogram with:

The Trapezoid (UK: Trapezium)
Trapezoid

Isosceles Trapezoid

It is called an Isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown.
And a trapezium (UK: trapezoid) is a quadrilateral with NO parallel sides:
Trapezoid  Trapezium  
US:  a pair of parallel sides  NO parallel sides 
UK:  NO parallel sides  a pair of parallel sides 
The Kite
... and that's it for the special quadrilaterals.
Irregular Quadrilaterals
The only regular quadrilateral is a square. So all other quadrilaterals are irregular.The "Family Tree" Chart
Quadrilateral definitions are inclusive.Example: a square is also a rectangle.
So we include a square in the definition of a rectangle.(We don't say "Having all 90° angles makes it a rectangle except when all sides are equal then it is a square.")
Using the chart below you can answer such questions as:
 Is a Square a type of Rectangle? (Yes)
 Is a Rectangle a type of Kite? (No)
Complex Quadrilaterals
Oh Yes! when two sides cross over, you call it a "Complex" or "SelfIntersecting" quadrilateral like these:
They still have 4 sides, but two sides cross over.
Polygon
A quadrilateral is a polygon. In fact it is a 4sided polygon, just like a triangle is a 3sided polygon, a pentagon is a 5sided polygon, and so on.Play with Them
Now that you know the different types, you can play with the Interactive Quadrilaterals.Other Names
A quadrilateral can sometimes be called: a Quadrangle ("four angles"), so it sounds like "triangle"
 a Tetragon ("four and polygon"), so it sounds like "pentagon", "hexagon", etc.
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