Quadrilateral ABCD is a rhombus and points C, D, and E are on the same line. Is quadrilateral ABDE a rhombus?

(1) The measure of angle BCD is 60 degrees.
(2) AE is parallel to BD



Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.



1) INSUFFICIENT:

Draw segment BD. Since BC = CD (because ABCD is a rhombus), Triangle BCD is an isosceles triangle. Since angle BCD = 60, the remaining angles in the triangle must also equal 60 degrees and Triangle BCD is actually an equilateral triangle. Since BD must also be equal to AB and AD, Triangle ABD is also an equilateral triangle and all of its angles measure 60 degrees. Since CD is parallel to AB and DE is an extension of line CD, we know that DE is also parallel to AB. Using AD as a transversal, we know the measure of angle ADE = 60. However, we know nothing about angles DAE or AED and with only one pair of opposite sides parallel we cannot conclude that quadrilateral ABDE is a rhombus.

(2) INSUFFICIENT:

Knowing that AE is parallel to BD allows us to conclude that alternate interior angles DAE and ADB are congruent. For the same reasons stated above, we know that DE is parallel to AB, and with two pairs of opposite sides parallel we know we have a parallelogram. But we have no further evidence that quadrilateral ABDE is a rhombus.

(1) and (2) SUFFICIENT:

If we combine both statements we can conclude that drawing segment BD creates three equilateral triangles. We already know ABDE is a parallelogram, and since all of its sides are congruent we can conclude that it is a rhombus.




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I think the explanation is incorrect. it should be (2) is SUFFICIENT
This is because DAE is equal to ADB. But also because DE is parallel to AB angle BAE is equal to andle ADE. Also since BA = AD (since ABCD is a rhombus) Hence triangle BAD is congruent to traingle ADE.
Therefore ABDE is a rhombus

shamikba,
You've done well with identifying angles that are equal. However, for ABDE to be a rhombus we also must know that all the sides must be equal. The problem is that we really don't know how long AB and DE are. Let's pretend that AE and BD are both 5. AB and DE could both be 5, or they could be 6. We really don't know. You can redraw the figure and lengthen AB and DE to help you see this.

Thank you,

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Jamie Nelson
ManhattanGMAT Instructor

N/a Deleted my response...

Okay! :-)

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Jamie Nelson
ManhattanGMAT Instructor

Sorry, I'm confused how a 3 equilateral triangles make a rhombus?

Thanks in advance.

Now i’m confused. Did someone say that three equilateral triangles made a rhombus?

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Tim Sanders
Manhattan GMAT Instructor