Triangle GAF is congruent to triangle EDF because they share the following congruent parts:
Angle AFB is congruent to angle BFD
Angle BFG is congruent to angle BFE
Angle GAF is congruent to angle EDF
Therefore, by the Angle-Angle-Side (AAS) Congruence Postulate, triangle GAF is congruent to triangle EDF.
The image you attached shows a geometry problem with the following givens:
Line segment BF bisects angle AFD
Angle BFG is congruent to angle BFE
Angle GAF is congruent to angle EDF
The problem asks us to prove that triangle GAF is congruent to triangle EDF.
We can prove this using the following steps:
Given:
Line segment BF bisects angle AFD
Angle BFG is congruent to angle BFE
Angle GAF is congruent to angle EDF
Prove:
Triangle GAF is congruent to triangle EDF
Proof:
Since line segment BF bisects angle AFD, we have:
Angle AFB is congruent to angle BFD
Since angle BFG is congruent to angle BFE, we have:
Angle BFG = angle BFE
Since angle GAF is congruent to angle EDF, we have:
Angle GAF = angle EDF
By the Angle-Angle-Side (AAS) Congruence Postulate, we have:
Triangle GAF is congruent to triangle EDF
Therefore, we have proven that triangle GAF is congruent to triangle EDF.