According to the $\text{Law of Sines}$, which set of angle measures could represent the angles of a triangle?

• A. $120°$, $30°$, $40°$
• B. $60°$, $60°$, $60°$
• C. $100°$, $50°$, $40°$
• D. $90°$, $90°$, $90°$

Answer: (C)

Verified step by step guidance

1. Recall that the sum of the interior angles of any triangle must be exactly \$180^\circ$. This is a fundamental property of triangles.
2. Check each given set of angles to see if their sum equals \$180^\circ\(. For example, for the set \)120^\circ\(, \)30^\circ\(, and \)40^\circ\(, calculate \)120 + 30 + 40$.
3. Eliminate any sets where the sum of the angles is not \$180^\circ$, since such sets cannot represent the angles of a triangle.
4. Understand that the Law of Sines states that for a triangle with angles \(A\), \(B\), and \(C\) and opposite sides \(a\), \(b\), and \(c\), the ratio \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) holds true, but this only applies if the angles form a valid triangle.
5. Confirm that the remaining sets of angles not only sum to \$180^\circ$ but also can satisfy the Law of Sines, which is always true for valid triangles, so the key step is verifying the angle sum.