And in this situation can math not work?

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Imagine you need to find an unknown like the following: using a Statement A you find a statement B (for example an equality) that let you find that unknown,but when you have statement B you dosen't always have statement A, which means Statements A and B are not equivalent,and then you find the unknown using statement B. Will that value be correct? for example: You need to find coordinates of a point D, with A,B,C given points that you have ABCD is parallelogram. You use the equality AB=DC (which is not equivalent to ABCD parallelogram) and find values of D. Will these values mean D is a parallelogram? and If not , how can I generally know how to get good values, does I have always to use equivalent statements?

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In general, if statement $A$ is given, and statement $A$ implies statement $B,$ and statement $B$ leads to a solution for the unique value or location of an unknown object, then you have solved the problem of finding the unknown assuming statement $A.$ The fact that you cannot form a chain of implications in the reverse direction is irrelevant.

An error that students often make is that they show only the implications in the reverse direction, that is, they start with the solution and show that it implies the given conditions, which is not a proof of any kind that the given conditions imply the solution.

In your particular example, you have a statement, "$ABCD$ is parallelogram," which implies another statement, "$AB=DC$." So far, this is fine. If the statement $AB=DC$ then allowed you to construct the unique location of the point $D$ with no other help, then you would have a valid construction. But the statement $AB=DC$ actually tells you only that $D$ lies somewhere on a circle whose center is $C$ and whose radius is equal to $AB.$ That is not enough information to know the coordinates of the unique point $D$ that makes $ABCD$ a parallelogram.

So in your example, your attempted construction fails, but the failure is for reasons completely different than the doubts you expressed in your question.