Question

could △abc be congruent to △adc by sss? explain.
yes, but only if \\(\overline{ab} \cong \overline{dc}\\).
yes, but only if \\(\overline{bc} \cong \overline{dc}\\).
no, because \\(\overline{ab}\\) is not congruent to \\(\overline{ac}\\).
no, because \\(\overline{ab} \cong \overline{da}\\).

Explanation:

Step1: Recall SSS Congruence

SSS (Side - Side - Side) congruence criterion states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Step2: Analyze Given Triangle Sides

In $\triangle ABC$ and $\triangle ADC$:

• We know that $AC = AC$ (common side, reflexive property).
• From the diagram, $AB\cong AD$ (marked with one tick mark) and we need to check the third sides. For SSS, we need $BC\cong DC$ as well.
• Looking at the options, the first option "Yes, but only if $\overline{BC}\cong\overline{DC}$" is correct because we already have $AB\cong AD$ and $AC\cong AC$, so we need $BC\cong DC$ to satisfy SSS. The other options are incorrect: the second option's condition ($AB\cong DC$) is not relevant for SSS here, the third option says $AB\cong DA$ which is already given (so it's not a reason for no), and the fourth option is wrong because $AB$ and $AC$ are not the sides we need to compare for SSS between these two triangles.

Answer:

Yes, but only if $\overline{BC}\cong\overline{DC}$ (the first option among the four given options)