Theory:

A four-sided closed two-dimensional shape is called a quadrilateral. It has four vertices, four sides and four angles.
Angle sum property of a quadrilateral
The sum of the angles of a quadrilateral is \(360º\).
 
Proof:
 
Given: \(ABCD\) is a quadrilateral.
 
Construction: Join \(BD\).
 
A1.png
 
Now, we divided the quadrilateral into two triangles \(BAD\) and \(BCD\).
 
\(\angle B = \angle ABD + \angle DBC\) - - - - - (I)
 
\(\angle D = \angle ADB + \angle BDC\) - - - - - (II)
 
We know that "sum of all the angles of a triangle is \(180^\circ\)".
 
In \(\Delta BAD\):
 
\(\angle DAB + \angle ABD + \angle BDA = 180^\circ\) - - - - (III)
 
Similarly, in \(\Delta BCD\),
 
\(\angle DBC + \angle BCD + \angle CDB = 180^\circ\) - - - - (IV)
 
Adding equations (III) and (IV), we get:
 
\(\angle DAB + \angle ABD + \angle BDA + \angle DBC + \angle BCD + \angle CDB = 180^\circ + 180^\circ\)
 
Rearrange the angles.
 
\(\angle DAB + (\angle ABD + \angle DBC) + \angle BCD + (\angle BDA + \angle CDB) = 360^\circ\)
 
\(\angle A + \angle B + \angle C + \angle D = 360^\circ\)  [using equations (I) and (II)]
 
That is, the sum of the angles of a quadrilateral is \(360^\circ\).
Types of quadrilateral
Name
Picture
Properties
Parallelogram
Parallelogram.png
1. Opposite sides are equal and parallel.
 
2. Opposite angles are equal.
 
3. Diagonals bisect each other.
Square
Square.png
1. All sides are equal and parallel.
 
2. All interior angles are \(90^\circ\).
 
3. Diagonals bisect each other at right angles.
Rectangle
Rectangle.png
 
1. Opposite sides are equal and parallel.
 
2. All interior angles are \(90^\circ\).
 
3. Diagonals bisect each other.
Rhombus
 Rhombus.png
1. All sides are equal.
 
2. Opposite angles are equal.
 
3. Diagonals are perpendicular.
Trapezium
 Trapezium.png
1. The bases of a trapezium are parallel.
 
2. Sum of adjacent angles on non-parallel sides are supplementary.
Kite
 Kite.png
1. Diagonals are perpendicular.
 
2. Diagonals bisect the vertex angles.
 
3. Non-vertex angles are congruent.
 
4. Two disjoint pairs of consecutive sides are congruent.
Properties of a quadrilateral
1. A square, rectangle and rhombus are all parallelograms.
 
2. A square is a rectangle and also a rhombus.
 
3. A rectangle or a rhombus is not a square.
 
4. A parallelogram is a trapezium, but a trapezium is not a parallelogram.
 
5. A kite is not a parallelogram.