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Statement I - Two similar triangles of equal area are congruent.<br>Statement II - $$ABCD$$ is a square. Area of the equilateral triangle having one of its sides as $$AC$$ is equal to $$\sqrt{2}$$ times the area of the equilateral triangle having one of its sides as $$AB$$.
Similar triangles with equal areas have side ratio $$\sqrt{1} = 1$$, so they are congruent and Statement I is true. In a square $$AC = \sqrt{2}\,AB$$, and the area of an equilateral triangle of side $$s$$ is $$\frac{\sqrt{3}}{4}s^2$$, so the ratio of the areas is $$(\sqrt{2})^2 = 2$$, not $$\sqrt{2}$$. Statement II is false.
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