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Suppose we have a regular triangular-base pyramid(A.K.A.: a Tetrahedron). Obviously every single triangular side on a tetrahedron "meets flush" with every single other side. Or, another way of putting it, the object can be constructed without holes or cuts. The question comes when I had to determine whether I could construct an irregular triangular-base pyramid using 4 clones of a given isosceles, scalene, or right triangle instead of an equilateral triangle. When I've tried to do this on paper using a net I could never fold it so that all the sides met uniformly without holes or cuts.

To see what I mean take the net of a tetrahedron. What if all triangles in the net were the same dimensions as each-other but irregular instead of equilateral? Could they be moved around so that when you folded them along their sides they'd produce an irregular pyramid? I don't feel like this is always possible. I'm quite bad at visualizing this problem myself(I swear I've tried) so I need some help. I feel like there's some fundamental geometry rule[s] that I'm missing here.

Both image links courtesy of Wikipedia: https://en.wikipedia.org/wiki/Tetrahedron

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Yes, one can construct a tetrahedron whose faces are all congruent, and those faces are not equilateral triangles. The key is to observe that each pair of opposite edges must be equal (in a tetrahedron $ABCD$ the edges $AB$ and $CD$ are opposite etc.).

As a simple example consider the tetrahedron with vertices $(\pm a, 0,b)$ and $(0,\pm a, -b)$ with $a$, $b>0$. Each face is an isosceles triangle with two edges $\sqrt{2a^2+4b^2}$ and one $2a$.

Wikipedia calls tetrahedra with congruent faces disphenoids. Not all triangles are possible as faces. For example, obtuse isosceles triangles.

Angina Seng
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