In a quadrilateral ABCD, E is a point of intersection of two diagonals

Question

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In a quadrilateral ABCD, E is a point of intersection of two diagonals AC and BD. Is ABCD parallelogram?

(1) AE = EC
(2) Angle ADC = Angle ABC

madgmat2019 · Answer

1) In quadrilaterals, 
the diagonals bisect each other in .....paralellogram, rectangle, rhombus, square....insuff
2)opp angles are equal in square, rectangle, rhombus, parallelogram...... Insufficient

So based on both 1,2 we cannot conclude whether ABCD is a parallelogram....

OA:E

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joohwangie · Answer

In a quadrilateral ABCD, E is a point of intersection of two diagonals AC and BD. Is ABCD parallelogram?

parallelogram have opposite sides of equal length/opposite angle are equal
(1) AE = EC
if AE = EC meaning BC = BA
but we still don't know the length of AD and CD
insufficient

(2) Angle ADC = Angle ABC
We know two opposite angle but doesn't know the other two angle/side.

together is sufficient.
we know that two different opposite sides and two different opposite angles are equal therefore it is a parallelogram.

C

exc4libur · Answer

(1) AE = EC: means the diagonals bisect each other, thus could be a para/kite, insufic.
(2) Angle ADC = Angle ABC: means the opposite angles are congruent, could be para/kite, insufic.

(1&2): since AE=EC then angles DAB=DCB, and since angles ADC = ABC, then BE=ED; 
in turn, this creates a rhombus, not a kite, and a rhombus is a parallelogram, sufic.

Answer (C)

lacktutor · Answer

In a quadrilateral ABCD, E is a point of intersection of two diagonals AC and BD. Is ABCD parallelogram?

Statement1: AE = EC
It could be a parallelogram or an isosceles trapezoid.
--> Trapezoid is not a parallelogram. Because one of the main properties of a parallelogram is that two pairs of opposite sides must be parallel to each other.
Insufficient

Statement2: Angle ADC = Angle ABC 
Yes, Opposite angles should be equal to each other in the parallelogram. 
--> But, we can see this property in a kite which is not a parallelogram. 
Insufficient.

Taken together 1 & 2,

AE = EC, 
Angle ADC = Angle ABC 
--> A kite will be changed to rhombus. Rhombus is a parallelogram.
Sufficient

The answer is C. 
 
Please, Find the attached file.

minustark · Answer

To be a parallelogram, the opposite sides and opposite angles have to be equal. Here, from Statement 1 , we can infer that one diagonal, BD bisects the other diagonal, AC. This can happen for a square too. NOT SUFFICIENT.

From Statement 2, it is known that two opposite angles are equal. Same as the previous information, this can happen in case of a square too. NOT SUFFICIENT.

Adding together, we are still confused whether ABCD is a square or parallelogram. NOT SUFFICIENT.

Answer is E.

exc4libur · Answer

I saw your diagram, but I don't understand how a kite will be changed to a rhombus.
Could you help me?  lacktutor

lacktutor · Answer

The diagonals of the kite are perpendicular to each other and DE= EB(only DB diagonal is divided into two equal halves) 
—> AE=EC, and Traingle BEC is right-angled triangle. 
 EC^{2} + EB^{2}=BC^{2} 
 AE^{2} + EB^{2}=AB^{2} 
—> BC=AB

—> the same thing in triangle CED and triangle AED

—> AB=AD=DC=CB

Hope it helps

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exc4libur · Answer

This part is where I am having trouble, how is DE=EB?

lacktutor · Answer

exc4libur , Statement2 says that We can see this feature either in a kite or in a parallelogram. —> Also, the diagonal which connect this angles is divided into two equal halves — It is the PROPERTY of a kite. Posted from my mobile device

CEdward · Answer

Hi all, I am really having a hard time dealing with these types of questions where the shape is ambiguous. What's your thought process like for how to approach?

In a quadrilateral ABCD, E is a point of intersection of two diagonals

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In a quadrilateral ABCD, E is a point of intersection of two diagonals

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