Triangles are congruent if all three sides in one triangle are congruent to the corresponding sides in the other.
When two triangles have corresponding sides with identical ratios, the triangles are similar.
Of course if triangles are congruent, they are also similar.
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Yes.
A triangle that can be made in more than one way
No. Equilateral triangles have 60o angles. There is no way to build a right-angle triangle with sides of equal length.
The way you do trigonometry is by working with triangles. Trigonometry is a math class that is taught in high school and usually followers algebra and geometry.
By definition all spheres are identical in shape because a sphere is perfectly round. That's what "sphere" means. In the same way, all perfect squares are identical and all equilateral triangles are identical. Spheres might be larger or smaller, but the shape of all is the same because there is only one kind of sphere.