I think the rubber band technique is effective. If you can stretch a side or dimension and come up with different results, then the information is INSUFFICIENT.

1. If BCD is 60 then BAD is also 60. Then we are left with two angles from left to right with 120 each. Imagine a straight line cutting the rhombus in half horizontally, what we got are two equilateral triangles ABD and BCD. For ABDE to become a rhombus, AE, BD,DE, and AE must have equal sides. Imagine pulling the line CDE a little longer through pt. E, then we could distort the figure and come up with a non-rhombus quadrialeteral. We could push it back and we could estimate a rhombus.

INSUFFICIENT.

2. Now imagine your rhombus ABCD and make it narrower, this will make BD and AE's lengths shorter than the size of a side of rhombus ABCD. Imagine your rhombus a little wider and this will make BD and AE's lengths longer. By rubber band technique, we know that we are not sure if ABDE is a rhombus.

INSUFFICIENT.

Together:
We know that ABD and BCD are equilateral triangles forming rhombus ABCD. Thus, line BD would be equal to all the sides of the rhombus.
Now we know that BD and AE are parallel each other fixed by the bordering lines of BA and CDE. Hence, BD = AE.
All the sides of the rhombus are equal to BD then also to AE.

To close the deal, AB and DE must be equal to become a rhombus. Since AE and BD are two parallel lines with equal length then, we are certain that AB and DE are also equal in length.

Answer: C