Is Quadrilateral ABCD a Parallelogram? GMAT Data Sufficiency

Question: Is quadrilateral ABCD a parallelogram?

1. Area of triangle ABC is equal to the area of triangle ACD
2. AD = DC

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 ABCD a parallelogram?”– 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:

In order to prove that a quadrilateral is a parallelogram atleast one of the following should be true:

1. Two pairs of opposite sides are equal and parallel
2. One pair of opposite side is equal and parallel
3. Both the pairs of the opposite angles are equal
4. Diagonals bisect each other

Considering the problem:

1. Area of triangle ABC = Area of triangle ACD
   From this we just know that this diagonal AC and on it’s two sides there are two triangles which have same area. If we would have given that the base of these two triangles is equal and parallel then we could prove that the quadrilateral is a parallelogram. But we can’t.
2. It just tells us that two adjacent are equal. This is of no use to us.
   Combining these two also doesn’t give us any concrete information to prove that the figure is a parallelogram.

So, this is insufficient.

Correct Answer: E

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