A parallelogram is a quadrilateral whose both pairs of opposite sides are parallel. 

Since ABEF is a parallelogram, so AF || BE and BF is the transversal.
$\therefore \angle AFB=\angle FBE=x$
$\text{[Alternate angles]}$
Also, since BCED is a parallelogram,
BE || CD and BD is the transversal.
$\therefore \angle CDB=\angle EDB=y$
$\text{[Alternate angles]}$ 
Now, 
$\angle FBD=\angle FBE+\angle EBD$ 
$\angle FBD=x+y$
And,
$\angle AFE=\angle BED$
$\text{[Corresponding angles]}$ 
$\therefore \angle AFE=z$ 
$\angle BED=\angle BCD=z$ 
$\text{[opposite angles of a parallelogram]}$
$\angle CDG+\angle BCD=\angle EBD+\angle BED$
$\text{[Sum of opposite interior angles is}\text{equal to exterior angle of a triangle]}$
$\Rightarrow \angle CDG=y+z-y$
$\Rightarrow \angle CDG=z$

Let the smallest angle of quadrilateral be $x^{\circ }$.
Then, the angles of the quadrilateral will be $x^{\circ }$, $2x^{\circ }$, $3x^{\circ }$ and $4x^{\circ }$.
Sum of the angles of a quadrilateral is ${360}^{\circ }$. 
$\Rightarrow {360}^{\circ }=x+2x+3x+4x$
$\Rightarrow 10x={360}^{\circ }$
$\therefore x={36}^{\circ },2x={72}^{\circ },3x={108}^{\circ }and4x={144}^{\circ }$

A parallelogram is a quadrilateral whose both pairs of opposite sides are equal and parallel. Since the pairs of opposite sides equal and parallel in a square, rhombus and rectangle,  they are all parallelograms.

By joining the mid points of the sides of a quadrilateral, another quadrilateral is formed whose opposite sides will be equal and parallel. Hence the quadrilateral formed is a parallelogram.

In a trapezium only one pair of opposite sides are parallel. 
In a parallelogram, both pairs of opposite sides are equal and parallel. 
Therefore a trapezium is not a parallelogram.

A figure formed by joining four points in an order is called a quadrilateral.
Consider the analogy of four people as four points.
In the first case, there are four collinear points which cannot form a quadrilateral.
In the second case, there are three collinear points which can form a triangle, but not a quadrilateral.
In the third case, the two pair of people face each other and thus form a quadrilateral.
In the fourth case, the two pair of people face exactly opposite but hold each others hands from behind and side and thus can for a quadrilateral.

• Trapezium cannot be a parallelogram.
   


• Rhombus is a parallelogram, whose sides are always equal.


• Rectangle is a parallelogram whose opposite sides are equal.


• Kite is not a parallelogram.

The angles of a quadrilateral are in the ratio 3:4:5:6.

Let the angles of the quadrilateral be $3x,4x,5x,6x$.

The sum of the angles of a quadrilateral is ${360}^{\circ }$.
$\Rightarrow 3x+4x+5x+6x={360}^{\circ }$
$\Rightarrow 18x={360}^{\circ }$
$\Rightarrow x={20}^{\circ }$

$\therefore 3x={60}^{\circ },4x={80}^{\circ },5x={100}^{\circ },6x={120}^{\circ }$