Which of the following statements is false?
$\mathrm{(A) \ All\ equilateral\ triangles\ are\ congruent\ to\ each\ other.}$
$\mathrm{(B) \  All\ equilateral\ triangles\ are\ convex.}$
$\mathrm{(C) \  All\ equilateral\ triangles\ are\ equianguilar.}$
$\mathrm{(D) \  All\ equilateral\ triangles\ are\ regular\ polygons.}$
$\mathrm{(E) \  All\ equilateral\ triangles\ are\ similar\ to\ each\ other.}$
An equilateral triangle is isosceles, and we find that $\angle A=\angle B=\angle C$ if we use the property of isosceles triangles that if two sides of a triangle are equal then the opposite angles are equal. Thus equilateral triangles are equiangular, and $C$ is true.
Regular polygons are both equilateral and equiangular, and so are equilateral triangles are both equilateral (by definition) and equiangular (by the above argument). Thus equilateral triangles are regular polygons and $D$ is true. 
Since all of the angles in an equilateral triangle are congruent, all equilateral triangles are similar by AAA similarity.  Thus, $E$ is true.
Since $\angle A=\angle B=\angle C$ and $\angle A+\angle B+\angle C=180$, $\angle A=60^{\circ}$. Since no other angles are above $180^{\circ}$, all equilateral triangles are convex.  Thus, $B$ is true.
This just leaves choice $\boxed{\mathrm{(A)}}$.  This is clearly false:  an equilateral triangle with side $1$ is not congruent to an equilateral triangle with side $2$.