Step-by-step explanation:

Given : TWO triangles ABC and DEF such that angle B = angle E , angle C = angle F and BC= EF

To Prove : triangle ABC congruent to triangle DEF

Proof : there are 3 possibilities

CASE 1

when AB= DE

in this case we have

AB=DE

angle B = angle E

and

BC= EF

so by SAS congruency rule triangle ABC congruent to triangle DEF

CASE 2

when AB is greater than ED

in this case take a point G on ED such that EG = AB.join GF.

NOW in triangles ABC & GEF

AB= GE

ANGLE B = ANGLE E

BC= EF

BY SAS CONGRYENCY RULE TRIANGLE ABC CONGRUENT TO TRIANGLE GEF

SO,

angle ACB = angle GFE

but,

angle ACB = angle DFE

therefore,angle GFE = angle DFE

this is possible only when ray FG COINCIDES with ray FD or G coincides with D. therefore AB must be EQUAL to DE .

In triangles ABC and DEF ,

AB=DE

ANGLE B = ANGLE E

BC= EF

SO, by SAS criteria of congruent triangle ABC = DEF

CASE 3

when AB is less than ED.

in this case take a point G on ED produced such that EG= AB. JOIN GF. Now proceeding exactly on the same lines as in CASE 2 we can prove that triangle ABC is congruent to triangle DEF.

Hence,in all the 3 cases we obtain triangle ABC congruent to triangle DEF..

PROVED