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ABCD is a quadrilateral with parallel sides. Is ABCD a rectangle?

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ABCD is a quadrilateral with parallel sides. Is ABCD a rectangle?  [#permalink]

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ABCD is a quadrilateral with parallel sides. Is ABCD a rectangle?

(1) AC and BD bisect each other.
(2) Sum of <DAB and <BCD is less than 180 degrees.

Originally posted by sahmedmartinez on 18 May 2016, 09:19.
Last edited by Vyshak on 18 May 2016, 21:17, edited 1 time in total.
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Re: ABCD is a quadrilateral with parallel sides. Is ABCD a rectangle?  [#permalink]

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New post 18 May 2016, 13:25
sahmedmartinez wrote:
ABCD is a quadrilateral with parallel sides. Is ABCD a rectangle?

(1) AC and BD bisect each other.
(2) Sum of <DAB and <BCD is less than 180 degrees.

Dear sahmedmartinez,
I'm happy to respond. :-)

I don't know the source of this question, but it seems to be expecting a slightly more formal understanding of geometry than the GMAT expects.

For example, in a formal geometry class, it would always be the case that if we named the quadrilateral ABCD, that this is the order of the vertices as we move continuously around the perimeter in one direction or another. In other words, if the top has A on the left and B on the right, and the bottom has C on the left and D on the right, then technically, by these standards, it would be illegal to call that quadrilateral ABCD---we would have to call it ABDC or something such as that. The GMAT always conforms to this rigorous standard in its own labeling, but it tends not to ask questions demanding that students understand this standard.

In this problem, we have to understand and apply this standard. In particular, we have to know that calling the quadrilateral ABCD means that A is opposite C and B is opposite D, so that AC and BD are diagonals, and that <DAB and <BCD are opposite angles.

Here's another geometry fact. There are four properties of parallelograms, which I like to call the BIG FOUR:
1) Opposite sides are parallel
2) Opposite sides are equal
3) Opposite angles are equal
4) Diagonals bisect each other

Any one of those is enough to guarantee that the shape is a parallelogram; thus, any one automatically implies the other three. I discuss this a little more in this blog article.


Now, with all that background, we can address the question.

Statement #1: AC and BD bisect each other.
If the diagonals bisect each other, then it is definitely a parallelogram and it definitely has all the other parallelogram properties. A rectangle is a special case of the parallelogram category, so ABCD could be a rectangle, but it doesn't have to be. We can give no definitive answer to the prompt question. This statement, alone and by itself, is not sufficient.

Statement #2: Sum of <DAB and <BCD is less than 180 degrees.
The sum of two opposite angles is less than 180 degrees. This is decisive. All the angles of rectangle, by definition, are right angles, 90 degree angles, so the sum of two opposite angles or any two angle in a rectangle would be equal to 180 degrees. If we can pick any two angles of the four, add them, and get anything other than 180 degrees, then the shape is most definitely not a rectangle. We can give a definitive "NO!" to the prompt question. Because we can arrive at a definitive answer, this statement is entirely sufficient all by itself.

First statement is not sufficient, but the second is. OA = (B)

Incidentally, that second statement is interesting. If the sum of opposite angles in a quadrilateral equals 180 degrees, then the quadrilateral could be rectangle, but that alone is not enough to guarantee it. It's possible for a completely irregular quadrilateral to have two opposite angles that equal 90 degrees.

Does all this make sense?
Mike :-)
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Re: ABCD is a quadrilateral with parallel sides. Is ABCD a rectangle?  [#permalink]

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New post 18 May 2016, 13:38
Thanks Mike,

So the reason I posted this question was because I came across it here:

f1gmat (dot) com / gmat-data-sufficiency-process-elimination#

The author concludes that statement 2 is not sufficient on its own, and his logic seemed flawed. I followed the same thought process as you; any sum of those two angles other than 180 is certainly no rectangle, and statement two is sufficient on its own. I've never come across an author use an example incorrectly, so I wanted to verify on this forum.

Thanks for the clarification.
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Re: ABCD is a quadrilateral with parallel sides. Is ABCD a rectangle?  [#permalink]

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New post 18 May 2016, 13:59
sahmedmartinez wrote:
Thanks Mike,

So the reason I posted this question was because I came across it here:

f1gmat (dot) com / gmat-data-sufficiency-process-elimination#

The author concludes that statement 2 is not sufficient on its own, and his logic seemed flawed. I followed the same thought process as you; any sum of those two angles other than 180 is certainly no rectangle, and statement two is sufficient on its own. I've never come across an author use an example incorrectly, so I wanted to verify on this forum.

Thanks for the clarification.

Dear sahmedmartinez,
Yes, my friend. You were perfectly 100% correct. What the folks have on that website is embarrassing. Good job spotting their mistake!
Mike :-)
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Re: ABCD is a quadrilateral with parallel sides. Is ABCD a rectangle?  [#permalink]

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New post 18 May 2016, 15:25
If we modify the original condition and the question, in order of ABCD to be a quadrilateral, we need to know the if A, B, C and D each has the same degree (A=degree, B=degree, C=degree and D=90). In the case of the condition 2), if A=(degree) and C<180, it cannot be that A=(degree) and C=90. Hence, the answer is no and the condition is sufficient. Hence, the correct answer is B.


-Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.
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Re: ABCD is a quadrilateral with parallel sides. Is ABCD a rectangle?  [#permalink]

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Re: ABCD is a quadrilateral with parallel sides. Is ABCD a rectangle? &nbs [#permalink] 27 Nov 2018, 10:45
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