Often while solving mensuration problems, I encounter situations where I need to establish the congruence of two quadrilaterals, mostly having been got all the four sides (and of course they are equal for both of them). This is where I get entangled.
[Assuming I have the sides in a particular order.] I have proved that when the quadrilaterals are a trapezium (but not a parallelogram), they are congruent. However, when it is not so, and I have other information such as some angles and diagonals. For example, visualizing a construction, I can see that having one diagonal gives us two cases, and thus not unique. But I am not able to prove that having two diagonals also gives us a unique case. With an angle or two opposite angles there are again two cases but I am not able to see anything else. I know that assuming that it a convex quadrilateral, simplifies the problem considerably, but it is not necessary.
So, my question is: Can anyone give me a characterization of when two quadrilaterals are congruent, knowing the sides and that they are equal? And how much does assuming that the quad is convex help us?
I have another doubt, let us suppose that the area of a type of quadrilateral is $f(a,b,c,d,e)$, where the letters are some elements of the quadrilateral. Then knowing those particular elements, make the quadrilateral unique? If this is true, this simplifies some problems immensely, such as the uniqueness of cyclic quadrilaterals given sides in order, which I have not been able to do otherwise.