Which of the following criteria always proves triangles congruent when using the $\mathrm{Law} \mathrm{of} \mathrm{Sines}$?

• A. $\mathrm{Angle}-\mathrm{Side}-\mathrm{Angle}$ (ASA)
• B. $\mathrm{Side}-\mathrm{Side}-\mathrm{Angle}$ (SSA)
• C. $\mathrm{Angle}-\mathrm{Angle}-\mathrm{Angle}$ (AAA)
• D. $\mathrm{Side}-\mathrm{Side}-\mathrm{Side}$ (SSS)

Answer: (A)

Verified step by step guidance

1. Recall that the Law of Sines relates the ratios of sides to the sines of their opposite angles in a triangle: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
2. Understand that to prove triangle congruence, the criteria must guarantee exactly one unique triangle can be formed with the given information.
3. Analyze each criterion: ASA (Angle-Side-Angle) specifies two angles and the included side, which uniquely determines a triangle.
4. Consider SSA (Side-Side-Angle), which can lead to the ambiguous case where zero, one, or two triangles may be formed, so it does not always prove congruence.
5. Note that AAA (Angle-Angle-Angle) only shows similarity, not congruence, and SSS (Side-Side-Side) is a valid congruence criterion but is not related to the Law of Sines directly.