#### Answer

$I$

#### Work Step by Step

Let's evaluate the four statements to see which one is false:
Option $F$ is true. All sides of an equilateral triangle are congruent.
Option $G$ is true. An equilateral triangle can be isosceles because in an isosceles triangle, at least two of the sides must be congruent.
Option $H$ is true. An equilateral triangle is also equiangular.
Option $I$ is false. If all angles of an equilateral triangle are congruent, then each angle must be a third of $180^{\circ}$, which is the exact sum of all the interior angles of any triangle. This means that each angle in an equilateral triangle can be exactly $60^{\circ}$, no more and no less.