Question

to prove that \\( \triangle def \cong \triangle dgf \\) by sas, what additional information is needed?
\\( \angle def \cong \angle dgf \\)
\\( \angle dfe \cong \angle dfg \\)
\\( \overline{de} \cong \overline{dg} \\)
\\( \overline{dg} \cong \overline{gf} \\)

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

For two triangles $\triangle DEF$ and $\triangle DGF$, we already have $DF = DF$ (common side) and $EF=GF$ (given from the markings in the figure).

Step2: Identify the required angle

By the SAS criterion, we need the included angle between the two pairs of sides. The sides are $EF, DF$ in $\triangle DEF$ and $GF, DF$ in $\triangle DGF$. The included angles are $\angle DFE$ and $\angle DFG$. So, if $\angle DFE\cong\angle DFG$, then $\triangle DEF\cong\triangle DGF$ by SAS.

Answer:

$\angle DFE\cong\angle DFG$