No, they are not.
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Yes.
All equiangular triangles are similar, but not congruent. They all have three 60 degree angles. This does not mean that they all have the same side lengths though. Therefore, they are only similar, not congruent.
No. Two regular hexagons are always similar to each other, but two random hexagons are not necessarily similar.
Their interior and exterior angles are always the same but their perimeters may vary in length. But that is what "similar" means; so the answer is YES.
always