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Bunuel
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I think this is a high-quality question and I agree with explanation.
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The existing explanations doesn't seem sufficient to me!

From statement 1,

We know that the diagram may be a rectangle/rhombus/square, hence insufficient.

Statement 2 also says the same since

AB+CD=BC+AD applies for all the three.

Is my reasoning correct?

Bunuel can you pls help me with this one!
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Darknight2
The existing explanations doesn't seem sufficient to me!

From statement 1,

We know that the diagram may be a rectangle/rhombus/square, hence insufficient.

Statement 2 also says the same since

AB+CD=BC+AD applies for all the three.

Is my reasoning correct?

Bunuel can you pls help me with this one!

No. All squares are rhombi. This is not the reason the answer is E. The point is that ABCD could be a kite, so NOT a rhombus.

Both kite and rhombus have the diagonals perpendicular to each other and the sum of the opposite sides equal to each other. So, ABCD can be a kite or a rhombus. Look at the diagram below:



Answer: E.
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Bunuel
Darknight2
The existing explanations doesn't seem sufficient to me!

From statement 1,

We know that the diagram may be a rectangle/rhombus/square, hence insufficient.

Statement 2 also says the same since

AB+CD=BC+AD applies for all the three.

Is my reasoning correct?

Bunuel can you pls help me with this one!

No. All squares and rectangle, are rhombi. This is not the reason the answer is E. The point is that ABCD could be a kite, so NOT a rhombus.

Both kite and rhombus have the diagonals perpendicular to each other and the sum of the opposite sides equal to each other. So, ABCD can be a kite or a rhombus. Look at the diagram below:



Answer: E.

For more practice Properties of Polygons Questions.
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Hi Bunuel

Is square part of Rhombus? If so, then if stm1 change to AC is perpendicular and bisect to BD, will answer change to C because it can only be square/ rhombus and not kite?

Thank you!
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Hi Bunuel

Is square part of Rhombus? If so, then if stm1 change to AC is perpendicular and bisect to BD, will answer change to C because it can only be square/ rhombus and not kite?

Thank you!

1. Yes, all squares are rhombi but not vise-versa.
2. No, even if we knew that AC bisects BD, the answer would still be E. Check the image here: https://gmatclub.com/forum/m11-183887.html#p1979291
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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