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It can be but it doesn't have to be. Some triangles are symmetrical and others aren't.
Not at all; pyramids could be either squashed down to a flattened version, or elongated to a taller version, and still be symmetrical.
No. But all isosceles triangles and equilateral triangles are.
Yes as would be the case in two similar equilateral triangles.
Yes. But not all isosceles triangles are equilateral.