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Math Expert V
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Question Stats: 59% (01:05) correct 41% (01:06) wrong based on 457 sessions

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

DS51602.01
OG2020 NEW QUESTION

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Bunuel wrote:

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

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

Cheers,
Brent

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The Logical approach to this question starts with the notion that the GMAC uses 5 different ways 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.

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The Logical approach to this question starts with the notion that the GMAC uses 5 different ways 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.

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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

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Bunuel wrote:

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

DS51602.01
OG2020 NEW QUESTION

Statement One Alone:

Adjacent sides PQ and QR have the same length.

Quadrilateral PQRS might or might not be a parallelogram. For example, if RS and SP also have the same length as PQ and QR (for example, PQ = QR = RS = SP = 4), then it’s a parallelogram (in fact, it’s a rhombus). On the other hand, if RS and SP don’t have the same length as PQ and QR (for example, PQ = QR = 4 and RS = SP = 5), then it’s not a parallelogram. Statement one alone is not sufficient.

Statement Two Alone:

Adjacent sides RS and SP have the same length.

Quadrilateral PQRS might or might not be a parallelogram. For example, if PQ and QR also have the same length as RS and SP (for example, RS = SP = PQ = QR = 4), then it’s a parallelogram (in fact, it’s a rhombus). On the other hand, if PQ and QR don’t have the same length as RS and SP (for example, RS = SP = 4 and PQ = QR = 5), then it’s not a parallelogram. Statement two alone is not sufficient.

Statements One and Two Together:

Even with two statements, quadrilateral PQRS might or might not be a parallelogram. If all 4 sides have the same length (for example, PQ = QR = RS = SP = 4), then it’s a parallelogram (in fact, it’s a rhombus). However, if PQ = QR = 4 and RS = SP = 5, then it’s not a parallelogram.

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