Final answer:
The additional piece of information needed to show the two triangles are congruent by SAS is FE=HE, which provides the second pair of equal sides, completing the side-angle-side requirement.
Step-by-step explanation:
To show that two triangles are congruent by the Side-Angle-Side (SAS) criterion, we need to know that two sides and the angle between them in one triangle are equal to two sides and the angle between them in another triangle. In this context, we are looking to identify the additional piece of information required to establish congruence between two triangles.
If we have two triangles, let's name them Δ1 and Δ2, and we know one pair of their sides are equal (GF=HF or GH=HF or GH=GF) and that they share a common angle, the additional information needed would be the length of another side that is included between known sides or angles. Since we already have GF=HF or GH=HF or GH=GF, the extra information required will be that another side is also equal in length (FE=HE) in order to use the SAS postulate for proving congruence.