Best Answer:
I understand... the first of these questions make absolutely no sense... I could understand if it were written "explain in which conditions we could have ..." because we don't have all the information necessary for those and could work on extra variables.

For what it's worth, here's what I would put:

a) ABCD is a parallelogram if and only if E = C.

b) BEFC is indeed a parallelogram, because we know that BC = EF and BC || EF.

c) vector AC = vec(AB) + vec(BC). By definition, we have vec(AB) = vec(DE) and vec(BC) = vec(EF). Therefore, vec(DF) = vec(DE) + vec(EF) = vec(AB) + vec(BC) = vec(AC).

CF is the result of moving AD by vector AC = vector DF. So CF = and || AD.

d) Because AD || CF and AC || DF, ACFD is a parallelogram.

e) vector AC = vec(AB) + vec(BC). By definition, we have vec(AB) = vec(DE) and vec(BC) = vec(EF). Therefore, vec(DF) = vec(DE) + vec(EF) = vec(AB) + vec(BC) = vec(AC).

f) AB = DE, BC = EF, AC = DF. They are more than congruent, they are the same triangle, one of which has been moved by vec(AD) = vec(BE) = vec(CF)

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