## In the quadrilateral PQRS, side PS is parallel to side QR

Math questions from any Manhattan Prep GMAT Computer Adaptive Test.
ruben

### In the quadrilateral PQRS, side PS is parallel to side QR

Hi,

I want to thank you for the great job you guys do with the forum and the website!!

Ok, here is the question. The information given in the question, PS and QR are parallel and the answer n* 1 (PS=QR), don't seem to define strictly a parallelogram .In other words, would a square still have 2 parallel sides that are also equal?

ThankS!!!!

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

(1) PS = QR

(2) PQ = RS
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. The opposite sides of a parallelogram also have equal length.

(1) SUFFICIENT: 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."

(2) INSUFFICIENT: We know from the question stem that opposite sides PS and QR are parallel, but have no information about their respective lengths. This statement tells us that the opposite sides PQ and RS are equal in length, but we don’t know their respective angles; they might be parallel, or they might not be. According to the information given, PQRS could be a trapezoid with PS not equal to QR. On the other hand, PQRS could be a parallelogram with PS = QR. The answer to the question is uncertain.

### parallelogram or trapezoid

(1) PS = QR

PS=QR, means there are 2 parrallel lines of same length, so if you join there respective ends , those 2 lines are alos going to be parallel and equal length. This is so because we can prove that if we draw one diagnol, the 2 resulting triangles will be similar triangle (They have 2 equal sides and the angle between them equal).

(2) PQ = RS

This can clearly be a trapezoid, but may or may not be parallelogram. Since PQ=RS, we can prove 2 sides to be equal and one angle to be equal but not the angle between 2 equal sides, so triangles cannot be similar.

Hope it helps, since it can be completely clear with diagrams which I can't draw here.
StaceyKoprince
ManhattanGMAT Staff

Posts: 8854
Joined: Wed Oct 19, 2005 9:05 am
Location: San Francisco

Hey, ruben, thanks for the vote of confidence. You list this problem as "GMAT Test 1" - we actually need the name of the author / source for copyright reasons before instructors can comment. Can you tell me where this is from? (I'm guessing it's one of ours, but I have to make sure.) Thanks!
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT
Guest

Stacey,

It is one of the MGAMT test. I will make sure to say where the questions come from from now on!

Thanks

Ruben
dbernst
ManhattanGMAT Staff

Posts: 300
Joined: Mon May 09, 2005 9:03 am

Ruben, let me know whether I am interpreting your original question correctly. I think you are questioning whether Statement (1) alone is sufficient since the figure could also be a square. If this is the case, the answer is that a square IS a parallelogram - it is simply a special type of parallelogram. Just as a parallelogram is a special type of quadrilateral (4 sided figure), both rectangles and squares are special types of parallelograms (and, of course, special types of quadrilaterals!).

-dan
goelmohit2002
Students

Posts: 226
Joined: Sat Jul 04, 2009 8:40 am

### Re: In the quadrilateral PQRS, side PS is parallel to side QR

OE said = (1) SUFFICIENT: 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 ?
Ben Ku
ManhattanGMAT Staff

Posts: 817
Joined: Sat Nov 03, 2007 7:49 pm

### Re: In the quadrilateral PQRS, side PS is parallel to side QR

A parallelogram is any quadrilateral where both pairs of opposite sides are parallel (and turns out that they also have equal length).

There are many different kinds of parallelograms:
- A parallelogram where all four sides have the same length is called a rhombus.
- A parallelogram where all four angles are right angles is called a rectangle.
- A parallelogram where all four sides are the same length, and all four angles are right angles is called a square.
So square is one kind of parallelogram.

Now, examining the question you pose, if PS is parallel to QR, and they are the same length, then we can determine that the other two sides (PQ and RS) are also parallel and the same length. You can try to draw two parallel lines with the same length, and you see that when you connect them, you'll always get a parallelogram.

The geometric proof uses a diagonal (for example, PR).
- Knowing that given parallel lines, alternate interior angles have the same measure, you know SPR and PRQ have the same measure.
- Comparing triangles SPR and PRQ, we know that PR in triangle SPR is congruent to PR in triangle PRQ, since it's the same segment. We know that PS and QR are congruent (given in the statement).
- Triangles SPR and PRQ are congruent (using SAS). Because the triangles are congruent, the sides PQ and RS are parallel and have the same length.

I hope that helps.
Ben Ku
Instructor
ManhattanGMAT
goelmohit2002
Students

Posts: 226
Joined: Sat Jul 04, 2009 8:40 am

### Re: In the quadrilateral PQRS, side PS is parallel to side QR

bku wrote:A parallelogram is any quadrilateral where both pairs of opposite sides are parallel (and turns out that they also have equal length).

There are many different kinds of parallelograms:
- A parallelogram where all four sides have the same length is called a rhombus.
- A parallelogram where all four angles are right angles is called a rectangle.
- A parallelogram where all four sides are the same length, and all four angles are right angles is called a square.
So square is one kind of parallelogram.

Now, examining the question you pose, if PS is parallel to QR, and they are the same length, then we can determine that the other two sides (PQ and RS) are also parallel and the same length. You can try to draw two parallel lines with the same length, and you see that when you connect them, you'll always get a parallelogram.

The geometric proof uses a diagonal (for example, PR).
- Knowing that given parallel lines, alternate interior angles have the same measure, you know SPR and PRQ have the same measure.
- Comparing triangles SPR and PRQ, we know that PR in triangle SPR is congruent to PR in triangle PRQ, since it's the same segment. We know that PS and QR are congruent (given in the statement).
- Triangles SPR and PRQ are congruent (using SAS). Because the triangles are congruent, the sides PQ and RS are parallel and have the same length.

I hope that helps.

Thanks a lot Ben !
Ben Ku
ManhattanGMAT Staff

Posts: 817
Joined: Sat Nov 03, 2007 7:49 pm

### Re: In the quadrilateral PQRS, side PS is parallel to side QR

you're welcome!
Ben Ku
Instructor
ManhattanGMAT
gorav.s
Course Students

Posts: 25
Joined: Wed Dec 26, 2007 12:05 pm

### Re: In the quadrilateral PQRS, side PS is parallel to side QR

This is easy and clear guys -
statement 1 is sufficient. Here you know both parallel sides are also equal . This is a necessary condition to declare it a parallelogram.

Statement 2, - question just says two lines PS and QR are parallel .
Assume one line PS is bigger and other line QR is smaller.
now still RS = PQ can happen and figue will not be a parallelogram but a trapezium.

I cannot draw the fig here , but vizualize it and you will understand
Ben Ku
ManhattanGMAT Staff

Posts: 817
Joined: Sat Nov 03, 2007 7:49 pm

### Re: In the quadrilateral PQRS, side PS is parallel to side QR

gorav.s, thanks for your response. I think that often questions that may seem to be "easy and clear" can be confusing (especially under the test-taking pressures) if the foundations in defining these geometric terms are not there. So the key is to be absolutely confident about the terms and their related properties.
Ben Ku
Instructor
ManhattanGMAT
its4christian
Course Students

Posts: 10
Joined: Sun May 10, 2009 5:54 am

### Re: In the quadrilateral PQRS, side PS is parallel to side QR

hi all,
gorav.s says that statement 2 is stating the PS and QR are parallel, but doesn't it says that PS and QR have the same length?

(2) PQ = RS

If that's the case then you cannot assume PS will be bigger than QR, so it should be a sufficient statment as well, correct?

Thank you
Chris
Ben Ku
ManhattanGMAT Staff

Posts: 817
Joined: Sat Nov 03, 2007 7:49 pm

### Re: In the quadrilateral PQRS, side PS is parallel to side QR

its4christian wrote:hi all,
gorav.s says that statement 2 is stating the PS and QR are parallel, but doesn't it says that PS and QR have the same length?

(2) PQ = RS

If that's the case then you cannot assume PS will be bigger than QR, so it should be a sufficient statment as well, correct?

Thank you
Chris

It is possible for PS to be bigger than QR, if PQRS is an isosceles trapezoid (the bases are PS and QR, the legs are PQ and RS).
Ben Ku
Instructor
ManhattanGMAT
its4christian
Course Students

Posts: 10
Joined: Sun May 10, 2009 5:54 am

### Re: In the quadrilateral PQRS, side PS is parallel to side QR

I got it! Thank you Ben!

Christian
Ben Ku
ManhattanGMAT Staff

Posts: 817
Joined: Sat Nov 03, 2007 7:49 pm

you're welcome!
Ben Ku
Instructor
ManhattanGMAT