Which of the following pairs of triangles can be proven congruent using the $\mathrm{Law} \mathrm{of} \mathrm{Sines}$?

• A. Two triangles with two sides and the included angle known ($\mathrm{SAS}$ case)
• B. Two right triangles with the hypotenuse and one leg known ($\mathrm{HL}$ case)
• C. Two triangles with all three sides known ($\mathrm{SSS}$ case)
• D. Two triangles with two angles and the side opposite one of them known ($\mathrm{AAS}$ or $\mathrm{ASA}$ cases)

Answer: (D)

Verified step by step guidance

1. Recall that the Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where \(a\), \(b\), and \(c\) are sides opposite angles \(A\), \(B\), and \(C\) respectively.
2. Understand that the Law of Sines is particularly useful when you know two angles and one side (AAS or ASA cases), because you can find the unknown sides by setting up ratios of sides to sines of their opposite angles.
3. Analyze the given cases: SAS (two sides and included angle), HL (right triangles with hypotenuse and one leg), and SSS (all three sides) are typically proven congruent by other methods (SAS, HL, SSS respectively), not by the Law of Sines.
4. Recognize that the Law of Sines helps prove congruence when you have two angles and a side opposite one of them (AAS or ASA), because knowing two angles determines the third, and the side opposite a known angle allows you to find the other sides using the Law of Sines.
5. Therefore, the pair of triangles that can be proven congruent using the Law of Sines is the one with two angles and the side opposite one of them known (AAS or ASA cases).