Final answer:
To prove that triangles HGE and FGE are congruent, one would utilize the Angle Bisector Theorem along with the Side-Angle-Side (SAS) Congruence Theorem, taking into account the proportional sides created by the bisecting segment GE and the congruent angles HGE and FGE.
Step-by-step explanation:
The student is looking for a proof that in a given geometric configuration, segment GE is an angle bisector of angle HEF and angle FGH, which leads to triangles HGE and FGE being congruent. To prove this, we would use the Angle Bisector Theorem, which states that an angle bisector in a triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides.
Since GE is given as the angle bisector for both angles, by the Angle Bisector Theorem, it divides the opposite sides proportionally. This means that the side ratios in triangle HGE and triangle FGE will be the same, and since they share segment GE and have an equal angle (the angle bisected by GE), the triangles will be congruent by the SAS (Side-Angle-Side) Congruence Theorem.
This results in △HGE ≅ △FGE due to the congruent shared side (GE), the congruent angles (angle HGE and angle FGE), and the proportional sides created by the bisecting of angles HEF and FGH.