in a rhombus sides are congruent (equal), diagonals are perpendicular

Is ABDE a rhombus-? for ABDE to be rhombus, the following conditions must be met AB=BD=DE=AE and AD perpendicular to BE.

st(1) \_BCD=60`, hence \_ADE=60` because of parallelism BC||AD. ABC=(180-60)`=120` and ABC=ADC. ABD=120`/2=60` and BDE=60+60=120`. Now if \_BDE=\_BAE, then ew have congruent sides BD=AE=AB=DE because triangles ABD=ADE are equilateral {ABD=120`/2=60` perpendicular diagonals dividing angles by 2 and forming 60-60-60` triangles}, BUT we don't know if \_BDE=\_BAE, Not Sufficient alone.
st(2) Not sufficient alone, as we don't know if all sides are congruent;

Combined st(1&2): must be sufficient as the sides are congruent.

IOM c

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: BCD = 60 degrees
In the figure below, ABDE is a rhombus:

In the figure below, ABDE is not a rhombus:

Insufficient.

Statement 2: AE||BD
In the figure below, ABDE is a rhombus:

In the figure below, ABDE is not a rhombus:

Insufficient.

Statements 1 and 2 together:
If BCD is 60 degrees, and AE||BD, then all the angles shown must be 60 degrees, all the triangles shown must be equilateral, and ABDE must be a rhombus:

Sufficient.

The correct answer is C.

GMATGuruNY wrote:
Statement 1: BCD = 60 degrees
In the figure below, ABDE is a rhombus:

In the figure below, ABDE is not a rhombus:

Insufficient.

Statement 2: AE||BD
In the figure below, ABDE is a rhombus:

In the figure below, ABDE is not a rhombus:

Insufficient.

Statements 1 and 2 together:
If BCD is 60 degrees, and AE||BD, then all the angles shown must be 60 degrees, all the triangles shown must be equilateral, and ABDE must be a rhombus:

Sufficient.

The correct answer is C.

Thanks GMATGuruNY for explaining it so well! You got a knack for simplifying problems. The solution makes sense now. I had gone with B but i understand my mistake now.

OA is C.