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805+ (Hard)|   Geometry|      
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Bunuel
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In a quadrilateral ABCD, E is a point of intersection of two diagonals 09 Oct 2019, 22:03
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C
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95% (hard)

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29% (01:57) correct 71% (01:34) wrong based on 85 sessions

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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
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C
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In a quadrilateral ABCD, E is a point of intersection of two diagonals 10 Oct 2019, 00:51
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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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In a quadrilateral ABCD, E is a point of intersection of two diagonals 10 Oct 2019, 02:57
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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
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In a quadrilateral ABCD, E is a point of intersection of two diagonals Updated on: 11 Oct 2019, 06:44
Originally posted by exc4libur on 10 Oct 2019, 04:54.
Last edited by exc4libur on 11 Oct 2019, 06:44, edited 1 time in total.
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Quote:
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
Show more

(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)
Attachments

main-qimg-5c9d73620e00b2058f712675de739d03.gif
main-qimg-5c9d73620e00b2058f712675de739d03.gif [ 16.25 KiB | Viewed 5251 times ]

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In a quadrilateral ABCD, E is a point of intersection of two diagonals 10 Oct 2019, 15:11
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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.
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Kite_rhombus.jpg
Kite_rhombus.jpg [ 115.84 KiB | Viewed 5133 times ]

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In a quadrilateral ABCD, E is a point of intersection of two diagonals 10 Oct 2019, 16:36
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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.
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In a quadrilateral ABCD, E is a point of intersection of two diagonals 11 Oct 2019, 05:25
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lacktutor
In a quadrilateral ABCD, E is a point of intersection of two diagonals AC and BD. Is ABCD parallelogram?

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

Show more

I saw your diagram, but I don't understand how a kite will be changed to a rhombus.
Could you help me? lacktutor
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In a quadrilateral ABCD, E is a point of intersection of two diagonals 11 Oct 2019, 06:03
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exc4libur
lacktutor
In a quadrilateral ABCD, E is a point of intersection of two diagonals AC and BD. Is ABCD parallelogram?

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


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

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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In a quadrilateral ABCD, E is a point of intersection of two diagonals 11 Oct 2019, 06:14
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lacktutor
exc4libur
lacktutor
In a quadrilateral ABCD, E is a point of intersection of two diagonals AC and BD. Is ABCD parallelogram?

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


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

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
Show more

This part is where I am having trouble, how is DE=EB?
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In a quadrilateral ABCD, E is a point of intersection of two diagonals 11 Oct 2019, 06:33
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exc4libur,

Statement2 says that <D and <B —opposite angles are equal to each other. —> 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.

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In a quadrilateral ABCD, E is a point of intersection of two diagonals 22 Mar 2021, 11:04
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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?
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