Question

to prove that δdef ≅ δdgf by sas, what additional information is needed?
○ ∠def ≅ ∠ dgf
○ ∠dfe ≅ ∠ dfg
○ (overline{de} cong overline{dg})
○ (overline{dg} cong overline{gf})

Explanation:

To prove \(\triangle DEF \cong \triangle DGF\) by SAS (Side - Angle - Side) congruence criterion, we need two sides and the included angle of one triangle to be congruent to the corresponding two sides and included angle of the other triangle.

1. First, from the diagram, we can see that \(DF\) is a common side, so \(DF\cong DF\) (reflexive property). Also, from the markings on the sides \(EF\) and \(GF\), we know that \(EF\cong GF\).
2. For the SAS criterion, we need the included angle between the two sides. The sides \(EF\) and \(DF\) form \(\angle DFE\) in \(\triangle DEF\), and the sides \(GF\) and \(DF\) form \(\angle DFG\) in \(\triangle DGF\). So we need \(\angle DFE\cong\angle DFG\) to satisfy the SAS criterion.

• Let's analyze the other options:
• Option \(\angle DEF\cong\angle DGF\): This would be related to AAS or ASA, not SAS, since it is not the included angle between the two known congruent sides.
• Option \(\overline{DE}\cong\overline{DG}\): If we use this with the other known side and the common side, it would be SSS (if we consider all three sides) or SSA (which is not a valid congruence criterion), not SAS.
• Option \(\overline{DG}\cong\overline{GF}\): This does not help in establishing the SAS criterion for \(\triangle DEF\) and \(\triangle DGF\) as it is not related to the sides and included angle of the two triangles in the required way.

Answer:

\(\boldsymbol{\angle DFE \cong \angle DFG}\) (the option with the text "\(\angle DFE \cong \angle DFG\)")