Hi, there. I'll throw in my 2¢.

Given ajith's crystal clear and 100% correct explanations, I'm surprised but the level of confusion in this post. I'll do what I can to clear things up.

Prompt: if ABCD is a quadrilateral, is AB = BC=CD=DA?

It's asking whether ABCD is an equilateral quadrilateral, that is to say, a rhombus.

It is not necessarily asking about a square, which is both equilateral and equiangular. That's a trap that the GMAT test-writer has laid for you. For a triangle, equilateral is automatically equiangular as well. For all higher polygons, quadrilaterals and up, equilateral and equiangular an independent of one another, and those shapes that happen to have both are quite special, quite rare, not to be taken for grant, not to be assumed willy-nilly.

Statement #1: AC is perpendicular to BD

The vertices of any quadrilateral are, by convention, named in continuous order as one goes clockwise or counterclockwise around the edge. Here, ABCD means that if we start at A, and walk around the edge, we will come to B, then C, then D, in that order.

A and C are not adjacent, but across from each other, so AC is a diagonal. BD is also a diagonal.

If we know only that the diagonals are perpendicular, the quadrilateral could be a square, or rhombus, or a kite, or an irregular quadrilateral. See p. 1 of the pdf for examples. This statement does not uniquely determine the quadrilateral, so it is insufficient.

Statement #2: AB + CD= BC + AD

This is an unusual constraint --- the sum of one pair of opposite sides equals the sum of the other pair of opposite side. Again, with this constraint alone, the quadrilateral could be a square, or rhombus, or a kite, or an irregular quadrilateral. See p. 2 of the pdf for examples. This statement does not uniquely determine the quadrilateral, so it is insufficient.

Combined Statements #1 & #2: AC is perpendicular to BD, and AB + CD = BC + AD

Here, I believe, no irregular quadrilaterals are possible any more. Nevertheless, as ajith has pointed out, both the kite and rhombus are possible here, so even here, it is not sufficient to determine that the quadrilateral is definitively a rhombus. Therefore, answer = E.

One important thing to keep in mind here are what I call the BIG 4 parallelogram properties:
(a) opposite sides parallel
(b) opposite sides congruent
(c) opposite angles congruent
(d) diagonals bisect each other

Any one of those is sufficient to prove that the quadrilateral is a parallelogram; any one of those four sufficient to prove the other three.

The only way you know you have sufficient information either to prove a quadrilateral is a parallelogram or to prove a quadrilateral is one of the "special" parallelograms (square, rectangle, rhombus) is if you have one of the BIG 4 in the mix some how. Here, even when Statements #1 & #2 are combined, it's not enough to prove one of the BIG 4, so not enough to guarantee the quadrilateral is a rhombus.

The difficulty of this question lies in confusing what could be true with what must be true.

Irregular quadrilaterals are legion, but a rhombus is one of the elite of the quadrilateral world.

The analog of this question:

Prompt: is some randomly chosen American the President of the US?
Statement #1: the randomly chosen American has two arms & two legs
Statement #2: the randomly chosen American's last name begins with a vowel

It's true that if the randomly chosen American were the President, Statements #1 and #2 would both be true, but that is NOT what GMAT DS is asking. GMAT DS is asking: if you know Statements #1 and #2 only, is that enough to guarantee, without a shadow of a doubt, that the randomly chosen American is the President? Obviously, if I tell you the person who lives next door to me has two arms, two legs, and last name that begins with a vowel, you are not going to leap to the conclusion that my next-door neighbor is the President. To leap to that conclusion would be supremely gullible.

Much in the same way, be hyper-reluctant to jump to the conclusion of "elite" quadrilateral (rectangle, rhombus, square) without sufficient evidence (which must include one of the BIG 4). Otherwise, that's just as gullible as leaping to the conclusion that my next-door neighbor is the President when given minimal information.

Does all this make sense? Please let me know if any of you have any questions at all about this.

Mike