Question

identifying additional information needed
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 \(EF = GF\) (given by the mark on the sides) and \(DF=DF\) (common side).

Step2: Analyze each option

• Option 1: \(\angle DEF\cong\angle DGF\) is not the included angle between the known sides.
• Option 2: \(\angle DFE\cong\angle DFG\) is the included angle between \(EF\) and \(DF\) in \(\triangle DEF\) and between \(GF\) and \(DF\) in \(\triangle DGF\).
• Option 3: \(\overline{DE}\cong\overline{DG}\) is not relevant to the SAS criterion with the given sides \(EF = GF\) and \(DF = DF\).
• Option 4: \(\overline{DG}\cong\overline{GF}\) is not relevant to the SAS criterion with the given sides \(EF = GF\) and \(DF = DF\).

Answer:

\(\angle DFE\cong\angle DFG\)