If one 2 side and include angle of two triangle are congruent how to prove that the both he triangle are congruent

Question

By Sas congruence

Answer 1

Triangles ABC and DEF

 

angle B = angle E

 

side AB = side DE and 

 

side BC = side EF

 

It is accepted by standard procedure that given the above the triangles ABC and DEF are congruent by SAS

 

However if further proof is needed consider the cosine formula; AC² = AB² + BC² -2*AB*BC * cosB ; applied to the triangle DEF; DF² = DE² + EF² - 2*DE*EF*cosE,  since AB=DE, and BC = EF and angles E and B are equal, then it stands to reason that AC = DF and would give the third side as being equal as well. Hence both triangles are congruent.