Question

practice using the sss and hl congruence theorems.
triangle def and triangle dgf are shown in the diagram
to prove that △def ≅ △dgf by sss, what additional information is needed?
∠def ≅ ∠dgf
∠dfe ≅ ∠dfg
overline{de} ≅ overline{dg}
overline{dg} ≅ overline{gf}

Explanation:

• The SSS (Side - Side - Side) congruence theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
• In \(\triangle DEF\) and \(\triangle DGF\), we already know that \(EF\cong GF\) (marked in the diagram) and \(DF\) is a common side (\(DF\cong DF\) by the reflexive property of congruence).
• To apply the SSS theorem, we need the third pair of sides to be congruent. That is \(DE\cong DG\).
• Options \(\angle DEF\cong\angle DGF\) and \(\angle DFE\cong\angle DFG\) involve angles, which are not relevant for the SSS criterion. Option \(\overline{DG}\cong\overline{GF}\) is not the required side - pair for the SSS congruence of \(\triangle DEF\) and \(\triangle DGF\) as we need the sides corresponding to the two triangles.

Answer:

\(\overline{DE}\cong\overline{DG}\) (i.e., the third option)