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.

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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.


transforming the original condition by variable approach method, we have 2 variables for the rhombus (side length and one diagonal side length). Since we need to match the number of variables and equations, we need 2 more equations. Since there is 1 equation each in 1) and 2), C is likely the answer and it turns out that C actually is the answer .



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Hi Mitch,
I didn't get how r u getting angle CBD=CDB=60DEG.Can u pl explain once more?

sandipgumtya wrote:Hi Mitch,
I didn't get how r u getting angle CBD=CDB=60DEG.Can u pl explain once more?

Angles inside a triangle must sum to 180.
Since statement 1 indicates that ∠BCD=60, and the angles inside ∆BCD must sum to 180, ∠CBD + ∠CDB = 120.

The prompt indicates that ABCD is a rhombus.
In a rhombus, all sides are EQUAL.
Thus, CB=CD.

If two sides of a triangle are equal, the opposite angles are also equal.
In ∆BCD, since CB=CD, ∠CBD = ∠CDB.

Since ∠CBD = ∠CDB, and ∠CBD + ∠CDB = 120, ∠CBD = ∠CDB = 60.

thanks mitch,got it.but what r the essential and sufficient conditions for a quadrilateral to be a rhombus?i didn't get the two figs in statement 2.