Is ABCD rectangular

That's not true. For example, consider isosceles trapezoid or kite with equal diagonals.

can you list the useful properties of kite. In GMATCLUB MATHBOOK also i have searched but am not able to get the properties of kite

1. not sufficient, could be a parallelogram
2. not sufficient, could be rhombus

1 & 2 combined -> ABCD is a square, which also fits the definition of a rectangle (all angles are 90°)
the answer is c

@mnazari . . not necessary a square. a rectangle also has equal diagonals.

@skamal7 . . properties of kite : the diagonals are prependicular

if kite is a parallelogram, then its also a rhombus . all sides are equal and diagonals are prependicular bisector of each other

Back to question

A says two adjacent angles are right. Not sufficient . the two remaning ones add upto 180 but may or may not be each 90

B says diagonals are equal. : may not even be a paralleogram

combined. . . two adjacent angles are 90 at B and C . . there are two diagonals BD and AC, these diagonals are equal. BC is surely a straight line . therefore since BD and AC are equal ( the two hypotenuse), BC is the same straight line, therefore the two heights AB and CD are also equal by pythagoras Theorem. the line joining AD will be parallel to BC. and the given shape is rectangle.

Draw a diagram to understand better. . I can add one if anyone can explain how to add it .

Is ABCD rectangular?

(1) Angle ABC and BCD are right angle

This tells us nothing about the other two angles. Insufficient.

(2) Two diagonal have the same length.

An iso. trapezoid has diagonals of the same length as does a square and rectangle. Insufficient.

1+2) Having two right angles plus identical diagonals means that the shape can be either a rectangle or a square. Either way ABCD is rectangular. Sufficient.

My answer is C.

Rephrasing the question: ABCD is retangular when ABCD is a parallelogram which has a right interior angle

(1): Angle ABC and BCD are right angles.
---> AB//CD. This proves that ACBD is a trapozoid with 2 right angles.
It does not say whether AD//BC. ---> NOT SUFFICIANT.


(2): two diagonal have a same length ---> ABCD could be a rhombus or a retangular ---> not sufficiant.

(1) and (2):
ABC is a right triangle at B ---> AC squared = AB squared + BC squared.
BCD is a right triangle at C ---> BD squared = BC squared + CD squared.

AC = BD ---> AB = CD.

---> ABCD is a trapozoid which has 2 equal bases and 2 right triangles
---> ABCD is a retangular.

OR i can prove:
AC = BD ---> compare 2 triangles ABD and BDC : ABD = BDC (side, angle, side)
---> AD = BC.
---> retangular
SUFFICIANT

I still don't get how can we get a figure with exactly two right angles with the restrictions given in Statement 1. Could anybody show such picture or at least describe it for one to reproduce?

Thanks!
Cheers
J

I'm not done yet. 2 months to go

Here it is:
Hope it helps.

[b][/b]Is ABCD rectangular?

(1) Angle ABC and BCD are right angle

This tells us nothing about the other two angles. Insufficient.

(2) Two diagonal have the same length.

An iso. trapezoid has diagonals of the same length as does a square and rectangle. Insufficient.

1+2) Having two right angles plus identical diagonals means that the shape can be either a rectangle or a square. Either way ABCD is rectangular. Sufficient.

------------------------------
I have one question here, 1+2 gives us Square or Rectangle. If all sides are equal then it will be square not Rectangle. If all sides are not equal then it will rectangle.
So here we are getting YES or NO . leading to E .. Please explain where I am missing.

Hi Schittuluri,
Even being square could be called rectangular.So, these two statements together suffice to say whether ABCD is rectangular or not.
Hope it clears.

Press Kudos if you wish to appreciate

The best approach to tackle statement questions in DS is as follows:

Step 1: Convert all the alphabetical statements in algebraic statements
Step 2: Reduce the number of variable to minimum
Step 3: Check how many variables are left. You may probably need that many statements to solve the questions but you might need lesser number of statements to answer the question.

Caution: Don't waste your time in solving the question. You have to analyse the data sufficiency and not solve the question.

Lets solve this using the approach given above:

Step 1: Is ABCD rectangular? can be reduced to the fact that angle ABC, BCD, CDA, and DAB are all right angles. If we deduce this, we can be assured that it is a reactangle.

Step 2: If 3 angles are 90 degrees, the fourth angle automatically becomes 90 degree. Hence, we need to know only 3 angles. Further look, even a group of opposite angle if each 90 degree, even then we can say it is a rectangle.

Step 3:

(1) Angle ABC and BCD are right angle.

ABC and BCD are consecutive angles, and not opposite angles, thus, we can't say about other two angles. Hence, it is not sufficient.

(2) Two diagonal have the same length

This statement does not tell anything about angles of the quadrilateral and hence not sufficient.

Now, if we take both the statements together, we know that one diagonal is AC which is a hypotenuse of AB and BC. Now, the other diagonal BD = AC as per statement 2. Since angle BCD is also a right angle, CD = AB. Now if you join AD, we get an angle ADC = 90 degree. Now we know that three angles are 90 degree and hence it is a rectangle. Option C) therefore is the right answer.

A quadrilateral is also a rectangle if any 2 angles are 90 degree and any two opposite sides are equal. Using both 1) and 2) together, we can see that ABC and BCD are 90 degree and after this we get AB = CD. Hence we can say using both the statements, we can answer the question.

Hope it helps!!!

Please Kudos if you like the reply!!!

each alone is insufficient.
1+2 -> 2 near angles are 90 degrees...if to picture a figure with 2 angles of 90 degrees, and if to draw 2 diagonals so that the 2 are equal, we would see that AB must be equal to CD, and AB must be parallel to CD. since the segments AB and CD form 90 degrees angle with the side BC, the side AD would be parallel to BC, and thus form another 2 90 degrees angles...
i understand that what I just said is for some absurd..but if to imagine the figure, everything makes sense...

Here we have to assume that ABCD is quadrilateral..

The question stem should clearly state that!
Please reframe the question~
It should say ABCD is a quadrilateral.

Kite with equal diagonals is a rhombus right?

Hi Bunuel,

In this question, combining statement 1 and 2 we get that AB=DC, but we still cannot find out whether AB=BC or DC=BC. So that quadrilateral can be either a square or a rectangle.

Either case we do not get a clear yes, & on this forum, many have said that square is a special case of rectangle or is rectangular, but still rectangle is not square, hence shouldn't the answer be E?

All squares are rectangles. So, if ABCD is a square it's a rectangle too.

Hi Bunuel,

Here, you mean ∠BAD = ∠BCD, I believe.