Given two triangles, $△abc$ and $△adc$, can they be proven congruent by the Side-Side-Side (SSS) criterion? Choose the best explanation.

• A. Yes, because the Law of Sines guarantees congruence if two angles and a side are known.
• B. Yes, because any two triangles sharing two vertices are always congruent by SSS.
• C. No, because SSS requires all three corresponding sides to be equal, and without information about side $bc$ and $dc$, congruence cannot be established.
• D. No, because SSS can only be used for right triangles.

Answer: (C)

Verified step by step guidance

1. Identify the triangles involved: \( \triangle abc \) and \( \triangle adc \). Notice that they share vertices \( a \) and \( c \), but the other vertices differ (\( b \) and \( d \)).
2. Recall the Side-Side-Side (SSS) congruence criterion: two triangles are congruent by SSS if all three pairs of corresponding sides are equal in length.
3. Check which sides are known or shared: the sides \( ab \) and \( ac \) in the first triangle correspond to \( ad \) and \( ac \) in the second triangle. However, the side \( bc \) in the first triangle corresponds to \( dc \) in the second triangle, and there is no information given about these two sides.
4. Since the lengths of sides \( bc \) and \( dc \) are unknown or not stated to be equal, the SSS criterion cannot be applied to prove congruence between these two triangles.
5. Therefore, without all three pairs of corresponding sides known to be equal, the triangles cannot be proven congruent by the SSS criterion.