AbeNeedsAnswers wrote: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.

E

Source: Official Guide 2020

For the quadrilateral PQRS to be a parallelogram, the opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

Let's take each statement one by one.

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

The opposite sides may or may not be equal. Insufficient.

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

The opposite sides may or may not be equal. Insufficient.

(1) and (2) together

Still, the opposite sides may or may not be equal. Insufficient.

The correct answer: E

Hope this helps!

-Jay
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AbeNeedsAnswers wrote: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.

E

Source: Official Guide 2020

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

Answer: E

Cheers,
Brent

Hi All,

We're asked if quadrilateral PQRS is a parallelogram? This is a YES/NO question can be answered with a bit of logic (and a couple of drawings). By definition, a parallelogram must have 4 sides and each pair of 'opposite' sides must be parallel.

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.
Brent's explanation provides a nice example of the second shape.
Fact 1 is INSUFFICIENT

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, INSUFFICIENT

Final Answer: E

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