Is Quadrilateral PQRS a Parallelogram? GMAT Data Sufficiency

Question: Is quadrilateral PQRS a parallelogram?

(1) Adjacent sides PQ and QR have the same length.
(2) Adjacent sides RS and SP have the same length.

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2.  Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3.  BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient. 

“Is Quadrilateral PQRS a Parallelogram? (1) Adjacent Sides PQ and QR Have the Same Length. (2) Adjacent Sides RS and SP Have the Same Length GMAT Data Sufficiency” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation: 

Approach Solution 1:

Target question: Is quadrilateral PQRS a parallelogram?
If you recognize that each statement on its own is not sufficient, we can jump straight to . . .

Statements 1 and 2 combined
There are infinitely-many quadrilaterals that satisfy BOTH statements. Here are two:

Case a: PQRS could be a square.

Since a square is a type of parallelogram, the answer to the target question is YES, quadrilateral PQRS IS a parallelogram

Case b: PQRS could be kite-shaped.

In this case, the answer to the target question is NO, quadrilateral PQRS is NOT a parallelogram

Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Hence, the correct answer is E.

Approach Solution 2:

In the given question, it is asked if quadrilateral PQRS is a parallelogram? This is a YES/NO question can be answered with a bit of logic. By definition, a parallelogram must have 4 sides and each pair of 'opposite' sides must be parallel.

Let us consider every possibility:

Fact 1) Adjacent sides PQ and QR have the same length.

'Adjacent' sides refer to two sides that are next to one another (and meet at a point):
-A Square fits this description - and is a parallelogram, so the answer to the question is YES.
-Any other 4-sided shape with 2 equal sides that touch and 2 others sides that are NOT the same length as the first 2 - that's NOT a parallelogram, so the answer to the question is NO.
Fact 1 is insufficient.

Fact 2) Adjacent sides RS and SP have the same length.
Fact 2 essentially provides the same information that Fact 1 provides (but about the other 2 sides). The examples that fit Fact 1 also fit Fact 2 - and lead us to two different answers (one "YES" and one "NO").
Fact 2 is insufficient
Combined, we know...

1) Adjacent sides PQ and QR have the same length.
2) Adjacent sides RS and SP have the same length.

Even with both Facts, we can end up with shapes that are parallelograms or not, so the answer to the question is inconsistent.
Combined, it is also insufficient.
Hence, E is the correct answer.

Approach Solution 3:

The Logical approach to this question starts with the notion that 5 different ways are used to tell us that a certain quadrilateral is a parallelogram:
1. We're given two pairs of equal opposite sides.
2. We're given two pairs of parallel sides.
3. We're given one pair of sides that are equal and parallel.
4. We're given two pairs of equal opposite angles.
5. The question states 'In the figure above, ABCD is a parallelogram...". 

In this particular question it is the adjacent sides that are equal, which means that even if both statements are taken into account, all we know is that it is a deltoid. Can it be a parallelogram? Sure, if all 4 sides are equal (i.e. a rhombus), but it doesn't have to be.
Hence, the correct answer is E.

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