No, they do not.
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Well, this is a question that you can argue on. It all depends on how you divide the rhombus. There can be a triangle and a trapezoid. There can also be two triangles. The last one that I can think of right now are two rhombuses inside the original rhombus.
There are a great many different shapes that are in Geometry. There are squares, circles, triangles, rhombus', and hexagons for example.
A Rhombus. If it makes two equilateral triangles with a diagonal, the shape must have all four sides of equal length; thus it is a rhombus, possibly a square. As an equilateral triangle is made, the angle between two of the sides is 60o; which is not 90o, so it can't be a square, so it must be a rhombus.
The answer is no. It is true of triangles, but not necessarily of other polygons. A good counterexample is the Rhombus. You can define a Rhombus as a quadrilateral with 4 congruent ( equal) sides. However, only the opposite angles are equiangular, not all 4 angles. Picture, if you will, a very elongated Rhombus to easily see this. The only equiangular Rhombus is the Square. Many regular polygons are equiangular but not all.
No. But all isosceles triangles and equilateral triangles are.