Question: Is quadrilateral q a rhombus?

(1) Quadrilateral q has four equal sides
(2) Quadrilateral q has four equal angles

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.

Correct Answer: A
Solution and Explanation:
Approach Solution 1:
The problem statement asked to find out whether the quadrilateral ‘q’ is a rhombus.

Statement 1 alone: Quadrilateral q has four equal sides.
Since the statement says the quadrilateral ‘q’ has four equal sides, therefore it is a rhombus.
A rhombus is also said to be an equilateral parallelogram.
Moreover, all squares are rhombus, but the vice-versa is not correct.
Therefore the quadrilateral having four sides equal the rhombus.
Thus statement 1 is a sufficient condition for answering the question that is whether the quadrilateral ‘q’ is a rhombus.

Statement 2 alone: Quadrilateral q has four equal angles
According to the mathematical theorem, it is not only a quadrilateral that has all four angles equal. The rectangle also has all four angles equal, unless it is a square.
But there is not enough information to deduce whether the quadrilateral is a square or not. This is because all squares are rhombuses. As a result, if q is a square, then it is a rhombus.
But, due to lack of information, it is not a sufficient condition.
Therefore Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.}

Approach Solution 2:
The problem statement asked to find out whether the quadrilateral ‘q’ is a rhombus.

Statement 1: Quadrilateral q has four equal sides
A rhombus is a quadrilateral that has all its sides of equal length.
A square is also a quadrilateral with all sides of equal length and all the interior angles are 90 degrees.
Thus all the squares are rhombus, but the opposite is not correct.
Hence, Quadrilateral q is a rhombus with all sides of equal length.
Therefore, Statement 1 alone is sufficient.

Statement 2: Quadrilateral q has four equal angles
To justify this, we consider that unless the square is a rectangle, its angles are not equal.
But whether the quadrilateral is a rectangle or not cannot be deduced due to a lack of information.
Therefore it is proved that statement 2 has insufficient information.
Hence statement 1 alone is sufficient but statement 2 is not sufficient.

Approach Solution 3:
The problem statement asked to find out whether the quadrilateral ‘q’ is a rhombus.

Statement 1: Quadrilateral q has four equal sides
A square is a quadrilateral, and it has all its sides equal.
So it can be inferred that the square is a rhombus.
Thus the information provided by the first statement is sufficient to say that all square is a rhombus, but all rhombus is not a square.
Hence, it can be said that quadrilateral ‘q’ is a rhombus.
Therefore, statement one is sufficient.

Statement 2: Quadrilateral ‘q’ has all four angles equal.
Quadrilateral q could be a rectangle (which is not a square, thus not a rhombus) as well as a square, thus a rhombus. Hence, Not sufficient.
Hence statement 1 alone is sufficient but statement 2 is not sufficient.

“Is quadrilateral q a rhombus?”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “Kaplan GMAT Math Workbook”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions include 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.

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