Rahul@gurome wrote:Properties of a rhombus are:
(1) Opposite angles of a rhombus are equal in measure.
(2) All the 4 sides are equal in length and opposite sides are parallel.

(1) Join the diagonal BD. Then triangles ABD and BCD will be equilateral triangles with each of the angles equal to 60 degrees.
But we have no info on angle AED, whether AE is parallel to BD or not and also we don't know whether all sides are equal.So, (1) is NOT SUFFICIENT.

(2) AE is parallel to BD again doesn't give sufficient info to prove that ABDE is a rhombus.
So, (2) is NOT SUFFICIENT.

Combining (1) and (2), we get angle ABD = angle AED = 60 degrees (AD is the transversal to parallel lines, AE and BD). This implies opposite angles are equal. This implies that all the 4 sides of ABDE are equal (since ABD and ADE are equilateral triangles). Opposite sides are parallel, which is sufficient to prove that ABDE is a rhombus.

The correct answer is[spoiler] (C)[/spoiler].

Hello,

I was not clear on the following mentioned above:

But we have no info on angle AED, whether AE is parallel to BD or not and also we don't know whether all sides are equal.

esp. why we also need AE to be parallel to BD?

Also, I am not clear with how once we combine I and II we get angle ABD = angle AED = 60'. Can you please explain? Thanks a lot.

Best Regards,
Sri

gmattesttaker2 wrote:

Rahul@gurome wrote:Properties of a rhombus are:
(1) Opposite angles of a rhombus are equal in measure.
(2) All the 4 sides are equal in length and opposite sides are parallel.

(1) Join the diagonal BD. Then triangles ABD and BCD will be equilateral triangles with each of the angles equal to 60 degrees.
But we have no info on angle AED, whether AE is parallel to BD or not and also we don't know whether all sides are equal.So, (1) is NOT SUFFICIENT.

(2) AE is parallel to BD again doesn't give sufficient info to prove that ABDE is a rhombus.
So, (2) is NOT SUFFICIENT.

Combining (1) and (2), we get angle ABD = angle AED = 60 degrees (AD is the transversal to parallel lines, AE and BD). This implies opposite angles are equal. This implies that all the 4 sides of ABDE are equal (since ABD and ADE are equilateral triangles). Opposite sides are parallel, which is sufficient to prove that ABDE is a rhombus.

The correct answer is[spoiler] (C)[/spoiler].

Hello,

I was not clear on the following mentioned above:

But we have no info on angle AED, whether AE is parallel to BD or not and also we don't know whether all sides are equal.

esp. why we also need AE to be parallel to BD?

Also, I am not clear with how once we combine I and II we get angle ABD = angle AED = 60'. Can you please explain? Thanks a lot.

Best Regards,
Sri

esp. why we also need AE to be parallel to BD?

To prove something is a rhombus we need both of the following conditions:
1. opposite sides are parallel. Hence AE needs to be parallel to BD if we are to prove ABDE as a rhombus.
2. all sides are equal. If all sides are not equal, even if we have AE parallel to BD, we only have a parallelogram, and not a rhombus.

Also, I am not clear with how once we combine I and II we get angle ABD = angle AED = 60'. Can you please explain?
If BCD = 60, because ABCD is a rhombus, AB=BC=CD=DA. Hence in triangle CBD=CDB (if two sides are equal, their opposite angles are equal). But in triangle BCD, sum of all angles = 180. Since one angle is 60, and other two are equal, each of the other two must be 60. So all angles of BCD are 60. BCD is equilateral triangle. (2x+60= 180 => x=60).

Now opposite angles of a rhombus are equal. So BAD = 60. With the same reasoning as above all angles of triangle BAD =60 as well. So ABD = 60. Since AE and BD are parallel, AED=BDC=60.
That's how we combine to get ABD=AED=60.

gmattesttaker2 wrote:

Rahul@gurome wrote:Properties of a rhombus are:
(1) Opposite angles of a rhombus are equal in measure.
(2) All the 4 sides are equal in length and opposite sides are parallel.

(1) Join the diagonal BD. Then triangles ABD and BCD will be equilateral triangles with each of the angles equal to 60 degrees.
But we have no info on angle AED, whether AE is parallel to BD or not and also we don't know whether all sides are equal.So, (1) is NOT SUFFICIENT.

(2) AE is parallel to BD again doesn't give sufficient info to prove that ABDE is a rhombus.
So, (2) is NOT SUFFICIENT.

Combining (1) and (2), we get angle ABD = angle AED = 60 degrees (AD is the transversal to parallel lines, AE and BD). This implies opposite angles are equal. This implies that all the 4 sides of ABDE are equal (since ABD and ADE are equilateral triangles). Opposite sides are parallel, which is sufficient to prove that ABDE is a rhombus.

The correct answer is[spoiler] (C)[/spoiler].

Hello,

I was not clear on the following mentioned above:

But we have no info on angle AED, whether AE is parallel to BD or not and also we don't know whether all sides are equal.

esp. why we also need AE to be parallel to BD?

Also, I am not clear with how once we combine I and II we get angle ABD = angle AED = 60'. Can you please explain? Thanks a lot.

Best Regards,
Sri

I posted a visual solution here:

https://www.beatthegmat.com/rhombus-t71762.html

Hello EagleEye, Hello Mitch,

Thanks for the excellent explanation. It is clear now.
@Mitch, thanks for the excellent visuals. It was very helpful.

Thanks again.

Best Regards,
Sri