Question

using the sss congruence theorem
triangle def and triangle dgf are shown in the diagram.
to prove that $\triangle def \cong \triangle dgf$ by sss, what additional information is needed?
options: $\overline{dg} \cong \overline{gf}$, $\angle def \cong \angle dgf$, $\angle dfe \cong \angle dfg$, $de \cong dg$

Explanation:

Step1: Recall SSS Congruence

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.

For \(\triangle DEF\) and \(\triangle DGF\), we already know from the diagram that \(FE\cong FG\) (marked with the same tick marks) and \(DF\) is a common side, so \(DF\cong DF\) (by the Reflexive Property of Congruence).

Step2: Identify Missing Side

To apply SSS, we need the third pair of sides to be congruent. The sides of \(\triangle DEF\) are \(DE\), \(EF\), and \(DF\); the sides of \(\triangle DGF\) are \(DG\), \(GF\), and \(DF\). We already have \(EF\cong GF\) and \(DF\cong DF\), so we need \(DE\cong DG\) to satisfy the SSS condition.

Let's analyze the other options:

• Option \(\overline{DG}\cong\overline{GF}\): This does not help in proving the congruence of the two triangles as per SSS, since \(DG\) and \(GF\) are sides of the same triangle (\(\triangle DGF\)) in a non - relevant way for SSS between \(\triangle DEF\) and \(\triangle DGF\).
• Option \(\angle DEF\cong\angle DGF\): This is an angle - angle - side or angle - side - angle related, not SSS.
• Option \(\angle DFE\cong\angle DFG\): This is an angle, not a side, so it does not satisfy SSS.

Answer:

DE \(\cong\) DG (the option "DE \(\cong\) DG")