Question

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

Explanation:

Step1: Recall SAS Congruence

SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

In $\triangle DEF$ and $\triangle DGF$, we know that $EF\cong GF$ (from the markings in the diagram) and $DF$ is a common side, so $DF\cong DF$.

Step2: Analyze the included angle

For SAS, 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 SAS.

Let's analyze the other options:

• Option 1: $\angle DEF\cong\angle DGF$: This is not the included angle for the sides we have, so it doesn't help with SAS.
• Option 3: $\overline{DE}\cong\overline{DG}$: This would be for SSS or another congruence, not SAS with the given sides.
• Option 4: $\overline{DG}\cong\overline{GF}$: This doesn't relate to the sides we need for SAS between $\triangle DEF$ and $\triangle DGF$.

Answer:

$\angle DFE \cong \angle DFG$ (corresponding to the option $\boldsymbol{\angle DFE \cong \angle DFG}$)