Data Sufficiency

In the quadrilateral PQRS, side PS is parallel to side QR. Is PQRS a parallelogram?

(1) PS = QR

(2) PQ = RS

OA = A.....

[spoiler]OE said = (1) SUFFICIENT:[/spoiler] We know from the question stem that opposite sides PS and QR are parallel, while this statement tells us that they also have equal lengths. The opposite sides PQ and RS must also be parallel and equal in length. This is the definition of a parallelogram, so the answer to the question is “Yes.”

Can someone please why the highlighted(bold) one is correct ? How to prove the same ? How can we come to the highlighted conclusion from the statement 1 alone ?

Can someone please help ?

PS is parallel to side QR. It doesn't mean, PS = QR; (Check the attached image).

Statement 1:

(1) PS = QR

If PS = QR, then the connecting lines (ie, PQ and RS) are parallel and equal. So, the quadrilateral PQRS is a Parallolegram.

Statement 2:

(2) PQ = RS

Even if PQ = RS, what if the diagram looks like the one I attached? So, we can't clearly say whether PS = QR. So, Statement 2 is insufficient.

OA should be A.

umaa wrote:If PS = QR, then the connecting lines (ie, PQ and RS) are parallel and equal.

Thanks. Can you please tell the mathematical reason for the same...basically how to prove the same with the data provided...

Take, PS = QR = 5 and They both are parallel.

So, the distance between P and Q & P R and S are equal. Whatever angle it may be. Check the attached image.

If the angle is 90 degree = Its a rectangle. Rectangle is a parallelogram.

If the angle is anything but 90, both PS and QR equal and parallel. Also, both PQ and RS are equal and parallel.

You can't explain this is numbers.

umaa wrote: If the angle is anything but 90, both PS and QR equal and parallel. Also, both PQ and RS are equal and parallel.

You can't explain this is numbers.

Thanks.

But, actually I am not able to understand how can we deduce the highlighted above from the information provided.....why will they not be unequal or non parallel...any mathematical reason ?

In the question, they've given side PS is parallel to side QR. and Statement 1 says PS and QR are EQUAL.

Take PS = QR = 10.

Both are parallel.

Try to draw this in a paper. You'll get the point.

umaa wrote:Try to draw this in a paper. You'll get the point.

Sorry to drag this point a bit long....but if we see in the paper yes it looks....but how to prove it mathmatically....

since without mathematical reason we cannot say for sure that statement 1 is sufficient....

IMO there must be some mathematical reason for the same.....

Why don't you try plug-in various angles and see what is the length of PQ and RS.

umaa wrote:Why don't you try plug-in various angles and see what is the length of PQ and RS.

Thanks Umaa....

But as per my experience with DS questions.....

we should never ever try to rely on substitution/specific solutions.....until and unless forced to...as the last resort......the substitution method etc. have very high probability of errors especially because we are many times not able to think of corner cases....(especially in the stress time of exam)

That is the reason why I am refraining from doing this....and trying to learn the mathematics behind it.....majorly in order to learn property that probably I am missing due to which I am not able to prove that statement I alone is sufficient....

Its a kind of formula that in a quadrilateral, if 2 opposite sides are equal and parallel, then the remaining two opposite sides are equal and parallel.

Memorizing these kind of formulas would help us answer quickly.

By the way, thanks for the tip (If its a DS question, find the answer mathematically and answer the question)

umaa wrote:Its a kind of formula that in a quadrilateral, if 2 opposite sides are equal and parallel, then the remaining two opposite sides are equal and parallel.

Memorizing these kind of formulas would help us answer quickly.

By the way, thanks for the tip (If its a DS question, find the answer mathematically and answer the question)

Hi Umaa,

As per my understanding, the definition of parallelogram is that

"it is a quadrilateral which have both pair of opposite sides as parallel and equal"

Actually from statement 1 alone...i am really not able to reach to above definition....

Don't know what I am missing....

But yes by doing figures, values...we can intelligently guess that it has to be parallelogram(especially because statement 2 gives us enough hint)