Is quadrilateral ABCD a rectangle?

Question

(1) The diagonals (AC and BD) are perpendicular to each other
(2) The sum of any 3 angles in the quadrilateral is 270 degrees

*Kudos for all correct solutions

BrentGMATPrepNow · Accepted Answer

(2) The sum of  any  3 angles in the quadrilateral is 270 degrees
The key word here is ANY. 
Let's call the 4 angles a, b, c and d
If ANY 3 angles add to 270, we can write 4 possible equations. 
a + b + c = 270. This means that  angle d MUST EQUAL 90 degrees  (since all 4 angles must add to 360 degrees)
a + b + d = 270. This means that  angle c MUST EQUAL 90 degrees  (since all 4 angles must add to 360 degrees)
a + c + d = 270. This means that  angle b MUST EQUAL 90 degrees  (since all 4 angles must add to 360 degrees)
b + c + d = 270. This means that  angle a MUST EQUAL 90 degrees  (since all 4 angles must add to 360 degrees)
At this point, we shown that all 4 angles must equal 90 degrees.

Cheers,
Brent

BrentGMATPrepNow · Answer

Great question! In fact, it illustrates why I created the question in the first place.

A square is a type of rectangle. So, if the quadrilateral is a square, then we can also conclude that the quadrilateral is a rectangle.

From the Official Guide: 
   A parallelogram with right angles is a rectangle, and a rectangle with all sides of equal length is a square.

For more on this, here's a video:
 https://www.youtube.com/watch?v=uQ356PWGq98

Bunuel · Answer

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broall · Answer

(1) Insufficient. The diagonals of a rectangle could be perpendicular to each other (then it's a square) or not.
(2) The sum of 4 angles in the quadrilateral is 360. Hence, any of 4 angles in the quadrilater is 90 degree. So ABCD is a rectangle.

The answer is B.

Omkar.kamat · Answer

Please re look... When diagonals are parallel...it's a Rhombus...not a Rectangle.

Answer shud be A. If the diagonals are perpendicular it's not a Rectangle.sufficient.

B is not sufficient, as the sum doesn't give any clarity about which figure it is.

Correct me if wrong.

Omkar Kamat
When The Going Gets Tough, The Tough Gets Going !!

broall · Answer

Sorry but the diagonals of a quadrilateral will never be parallel.

Rhombus is a quadrilateral that its diagonals are perpendicular, not parallel, at the midpoint of each.

Also, a Rhombus could be a square, so it is a rectangle, but a rectangle isn't necessarily a Rhombus

A Rectangle is a quadrilateral with 4 right angles

Omkar.kamat · Answer

Sorry but the diagonals of a quadrilateral will never be parallel.

Rhombus is a quadrilateral that its diagonals are perpendicular, not parallel, at the midpoint of each.

Also, a Rhombus could be a square, so it is a rectangle, but a rectangle isn't necessarily a Rhombus 
Sorry. I meant perpendicular.

Omkar Kamat
When The Going Gets Tough, The Tough Gets Going !!

broall · Answer

Here is why A is insufficient Untitled.png

vikrantsharma725@gmail.com · Answer

A is clear that quadrilateral can be rhombus as diagonals of rhombus are also prependicular.

but why answer is B. as it can be square also since it is given in B that sum of any 3 sides is 270 degree. and it can be rectangle also. but if we combine B and A then diagonals of rectangle are prependicular and sum of 3 sides is also 270 degree.

so please tell me why not answer is C

broall · Answer

The answer is not C because we need only B enough to deduce that ABCD is a rectangle.

(1) is insufficient due to #7.

Let me make clearer in (1). If diagonals of ABCD are prependicular, ABCD could be a rhombus or not.
If ABCD is a rhombus, it could be a square or not.
If ABCD is a square, it's also a rectangle.

So (1) => ABCD could be a rectangle or not. Insufficient.

(2) A quadrilateral with 4 right angles is a rectangle. If ABCD is a square, it's clearly a rectangle since square is a special type of rectangle. In all cases, we have ABCD is a rectangle. So (2) is sufficient to answer the question.

Hope this helps.

vaibhav8644 · Answer

Perfect nguyendinhtuong :D

goalMBA1990 · Answer

I have a query for statement 2. We are told that sum of any 3 angles is 270. So how can we assume that each will be 90? It can be any value. So ans should be E. What am I missing?

Chandan17 · Answer

The sum of any 3 angles in the quadrilateral is 270 degrees.

It means the 4th angle will be 90 degree only. Try to understand it like this: If sum of angles a, b and c = 270 degree then angle d will be 90 degree. If sum of angles a, b and d = 270degree then angle c will be 90 degree and so on. So, it is a rectangle.

nemesiswv · Answer

is-quadrilateral-rstv-a-rectangle-130928 .Please check it.

Sum of angles T, S, R is 270 degree and angle V is 90 degree. Yet TSRV is not a rectangle. So correct me why the answer should not be E.

MSIMBA · Answer

B is correct

(1) is insufficient due to #7.

Let me make clearer in (1). If diagonals of ABCD are prependicular, ABCD could be a rhombus or not.
If ABCD is a rhombus, it could be a square or not.
If ABCD is a square, it's also a rectangle.

So (1) => ABCD could be a rectangle or not. Insufficient.

(2) A quadrilateral with 4 right angles is a rectangle. If ABCD is a square, it's clearly a rectangle since square is a special type of rectangle. In all cases, we have ABCD is a rectangle. So (2) is sufficient to answer the question.

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Is quadrilateral ABCD a rectangle?

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Is quadrilateral ABCD a rectangle?

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