I have not got this statement myself. So, kindly Prove this entire statement given in figure with the help of different instance. ( About equialgular triangle that, how two congruent triangle will be equiangular and two equiangular triangles need not to be congruent)

Two make 2 triangles congruent their angles and corresponding angles should be equal i.e.
ABC and DEF are congruent only if
AB = DE
BC = EF
CA = FD
and
angle A = D
angle B = E
angle C = F
Angles alone are not enough because  we can draw an equilateral triangle with side 6 and 8 cm, angles are equal but sides are NOT.

However we do know that we can draw unique triangles  if
SSS ( 3 sides)
SAS( 2 sides and included angle)
AAS( 2 angles and common side) are known 
Thus
if two triangles match w.r.t SSS or SAS or AAS they would be congruent too