Let us categorize quadrilaterals as parallelograms and non-parallelograms.
Then rhombi, rectangles and squares come under parallelograms.
A square can be broadly considered as a rhombus or as a rectangle.
Trapezoids and kites fall under non-parallelograms.
But the two have nothing in common (except having four sides and four angles which is true for any quadrilateral).
What is common to a rhombus and rectangle?
The pair of opposite sides are parallel.
What is common to rhombus and square ?
All the sides are equal and the diagonals are perpendicular.
What is common to square and rectangle ?
All the sides are equal and the angles are equal (to 90 °).
A quadrilateral is a figure having four sides.
A quadrilateral with a pair of parallel sides becomes a trapezoid.
(A trapezoid in which the non-parallel sides are equal is an isosceles trapezoid)
A trapezoid in which second pair of sides are also parallel becomes a parallelogram.
A parallelogram in which all the internal angles are 90° becomes a rectangle.
A rectangle in which all the sides are equal becomes a square.
A quadrilateral in which two pairs of adjacent sides are equal becomes a kite.
A kite in which all the sides are equal becomes a rhombus. (A parallelogram with equal sides is also a rhombus)
A rhombus (or kite) in which all the angles are equal becomes a square.