About a math proof concerning Parallelograms?

Question

There was a question that told me to:
draw a diagram
state the given
outline my proof
about prooving that a parallelogram is a rhombus ONLY IF the diagonals of it are perpendicular and the question did not say anything else (i think)
So i drew a parallelogram and points ABCD and E as the intersection of the diagonals. but the part im worried about is the "given" part because i wrote down AB is parallel to CD and BC is parallel to DA, and *AB = CD and CD = DA, and also that AE = CE and BE = DE.* Im worried about the fact that i put between the *'s because they aren't directly given in the question but i got them from the fact that the diagram is a parallelogram. My question is whether this is right
btw this is 9th grade high school math

Answer 1

It utilizes the same principals that a square is always a rectangle, but a rectangle is not always a square.

If the intersection of the diagonals are perpindicular, then the that insures that the sides of the parallelogram are equal, and thus a rhombus. Same thing for the diagonals of a rectangle...when they are perpindicular, it is a square. Hope this helps you out.

Answer 2

You should start by saying ABCD is a parallelogram with reason being "Given".

Then you can say AB || CD, BC || DA, etc with reason "Definition of parallelogram".

BTW, you are doing just fine. This was a good question.

Answer 3

Your statement *AB = CD and CD = DA, and also that AE = CE and BE = DE.* is right. However, you need to prove CD = DA based on AE = CE and BE = DE (diagonals of a parallelogram bisects each other) and the diagonals are perpendicular.

∆AED ≌ ∆CED (by HL)
CD = DA (by CPCTC)