I've been trying to find, without much success, 4 triangles whose corresponding sides are congruent that cannot be folded into a tetrahedron.
Anyone has any clue how to approach this problem?
I've been trying to find, without much success, 4 triangles whose corresponding sides are congruent that cannot be folded into a tetrahedron. Anyone has any clue how to approach this problem? 


What do you mean by "corresponding sides"? If what you mean that you have a gluing diagram which is consistent, just take your triangles $ABC, ABD, ACD, BCD$ in such a way that the angles at $A$ in all three triangles sharing that vertex is $5\pi/6,$ and otherwise the three triangles with vertex at $A$ are isosceles (so the other two angles are $\pi/10$) the triangle $BCD$ is equilateral. Notice that these triangles do not glue into a tetrahedron, since the total angle at $A$ is greater than $2\pi$ (since $15/6 > 2$). 


A tetrahedron with congruent faces will have all acute face angles. No obtuse or right angles. 

