Why do we say this for triangles?

Question

Case 1 ;Let us say there is a triangle $ABC$ and one $DEF$. If $AB = DE$ can we also say that $\angle BCA = \angle EFD $ ? What is the proof and reason for saying this ?

Case 2 : Supose $BC = EF$ along with $AB=DE$.

Answer 1

It is not true for either case. Let's say $BC = EF = AB=DE$ (Which is a particular case of your conditions). We can choose the angles $\angle ABC$ and $\angle DEF$ at will, so we can make $ABC$ a right triangle at B (meaning $\angle ABC = 90º$) and $\angle DEF$=60º.

If you make the drawing you can clearly see that the two triangles are not equal and that their angles $\angle BCA$ and $\angle EFD$ are diferent. $\angle BCA$ = 45º (since it's a right isosceles triangle) and $\angle EFD$ = 60º (since an isosceles triangle with an angle of $60º$ between two equal sides will be equilateral).