I don't understand the lecturer, solving the question, says that the rectangles are congruent each other, so the result can be obtained by proportioning them to each other. However, AFAIK two quadlaterals are congruent to each other such that sides an interior angles of one quadrilateral in corresponds with sides and angle of another and to prove that all corresponding pairs of sides and angles are congruent.
I think parallelism property may be used to proportion, but not directly because they are quadlateral.
A means the area.
Would you mind picturize the solution?
None of your rectangles are congruent to any other because congruence requires they be the same size. Still, from the fact that the area of a rectangle is the product of the length and width, you can say that rectangles that share one side have areas in proportion to the other side. So from $A(AEKH)=12, A(HKGD)=8$ you can conclude that $AH=\frac 32HD$. Similarly $EB=\frac 52AE$ and $A(KFCG)=\frac 52\cdot 8=20\ cm^2$