Is quadrilateral ABCD a parallelogram?

This one tricked all of you. (2) is not sufficient. ABCD could be a kite:




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Thanks Bunuel! for correcting it.

Is quadrilateral ABCD a parallelogram?

Stat1: All four internal angles of ABCD are equal.
if 4 internal angles are equal, then ABCD will be square or rectangle. In both cases, it will be a parallelogram. Sufficient

STat2: AC, a diagonal of ABCD, divides ABCD into two congruent triangles.
It can be KITE . So, it is a not parallelogram. Not Sufficient

So, I think A.

imo D

Is quadrilateral ABCD a parallelogram?

(1) All four internal angles of ABCD are equal.
ABCD will be square or rectangle , if 4 internal angles are equal .
.. we can confidently say that it will be a parallelogram. Sufficient


(2) AC, a diagonal of ABCD, divides ABCD into two congruent triangles.

=>
opposite sides are equal and parallel.
So we can confidently say that it is a parallelogram. Sufficient

Is quadrilateral ABCD a parallelogram?

Statement (1): All four internal angles of ABCD are equal.

ABCD will be a square or rectangle, if all the 4 internals angles are equal.
We can say that ABCD is a parallelogram.

The statement is Sufficient.

Statement (2): AC, a diagonal of ABCD, divides ABCD into two congruent triangles.

Opposite sides are equal and parallel. So, it is a parallelogram.

The statement is Sufficient.

Both Statement are sufficient, therefore the answer is D.

Or an isosceles trapezoid.

No, isosceles trapezoid does not meet any of the statement:



All four internal angles of ABCD are NOT equal and AC does NOT divide ABCD into two congruent triangles.


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For 1) If all angles are equal i.e. each angle is 90 degrees. Hence figure is either a rectangle or a square.

2) as bunnel said, can be a kite or a trapezium with a pair of non parallel sides equal

Hi devsawla,


Bunuel has rightly said. However, you have wrongly interpreted it.
For statement 2 (AC, a diagonal of ABCD, divides ABCD into two congruent triangles), a trapezium / trapezoid or an isosceles trapezium / trapezoid will never be a case.

Because if we see the content and figure in the link:
https://gmatclub.com/forum/is-quadrilat ... l#p2742986

It clearly states that isosceles trapezium / trapezoid (or even a trapezium / trapezoid) does not meet any of the two statements. And we can see that. Please let me know if you are still not clear on this.

The cases that satisfy the requirements for statement 2 are: a kite, a parallelogram, a rhombus, a rectangle and a square. So, Statement 2 alone is not sufficient because of the possibility of a kite case. All other cases (except kite) are a parallelogram.


Hi Bunuel,
Please correct my understanding or if I have interpreted wrongly in this post/reply.

Regards,
Ravish.

Thanks for pointing it out.
I do see how I was incorrect

Posted from my mobile device

Hi TarunKumar1234,

You have written for statement 2: "It can be KITE . So, it is a not parallelogram. Not Sufficient".
There are two fundamental flaws here.
The first one: If only a kite satisfies the statement 2 criteria, then only 1 case is possible and it's not a parallelogram. Hence, the statement 2 alone should be sufficient to say that it's not a parallelogram.

However, the cases that satisfy the requirements for statement 2 are: a kite, a parallelogram, a rhombus, a rectangle and a square. So, statement 2 alone is not sufficient because of the possibility of a kite case. All other cases (except kite) are a parallelogram.
So, statement 2: can be a parallelogram or cannot be a parallelogram. Statement 2 alone is not sufficient.

Second flaw is that statement 1 and statement 2 can never contradict each other. Is quadrilateral ABCD a parallelogram?
As per statement 1: Yes, it will be a parallelogram. Statement 1 alone is sufficient to say that quadrilateral ABCD is a parallelogram.
As per statement 2: (As you have written: "It can be KITE. So, it is a not parallelogram. Not Sufficient" Yes, it will NOTbe a parallelogram. Statement 2 alone is sufficient to say that quadrilateral ABCD is NOT a parallelogram.
Here, statement 1 and statement 2 contradict each other. Hence, some logical flaw or something we missed while solving each statement independently.

Hi Bunuel,
Please correct my understanding or if I have interpreted wrongly in this post/reply.

Regards,
Ravish.

ravish
I mentioned, It can be kite. Hope you observed that Parallelogram is easy to see from Stat2. So, It is not sufficient.
So, stat1 is sufficient. Hope, it is clear.

Bunuel To be clear:

Is this accurate?

-diagonals of a square, rectangle, rhombus, and parallelogram will divide the polygon into two congruent triangles
-square: 45-45-90
-rectangle: variable depending on ratio of sides
-rhombus: 60-60-60 (equilateral)
-parallelogram: What does this get you?
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Text in red is not correct. It's true only for a special rhombus which has adjacent interior angles equal to 60° and 120°. The diagonal connecting 120° degree angles divides the rhombus into two equilateral triangles.



As for parallelogram, each diagonal divides the quadrilateral into two congruent triangles.


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