I am not completely brushed up on all the geometric proofs. But in that problem, where does it tell you that GJ is an angle bisector of HGK? You'd first have to prove that GJ is an angle bisector of HGK given that HJ and JK are congruent.
Another way to prove that those two triangles are equal is to prove SAS and thus they are congruent.
Prove the small triangles are congruent from given info SAS or SSS.
Prove angle HMJ congruent KMJ - corresponding parts of congruent triangles are congruent.
Prove HMG and KMG are = 180 and congruent - supplementary angles to HMJ and KMJ
MH congruent MK - corresponding parts of congruent triangles are congruent.
MG congruent MG - reflexive.
The triangles are congruent by SAS.
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