Which of the following combinations of measurements could form a triangle according to the $Law of Sines$?

• A. Angle A = $80°$, Angle B = $70°$, Side a = $3$
• B. Angle A = $100°$, Angle B = $90°$, Side a = $7$
• C. Angle A = $120°$, Angle B = $70°$, Side a = $4$
• D. Angle A = $30°$, Angle B = $60°$, Side a = $5$

Answer: (D)

Verified step by step guidance

1. First, 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. For each given set of measurements, calculate the third angle \(C\) using the triangle angle sum property: \(C = 180^\circ - A - B\).
3. Check if the third angle \(C\) is positive and less than \$180^\circ$; if not, the measurements cannot form a triangle.
4. Using the Law of Sines, find side \(b\) corresponding to angle \(B\) by rearranging the formula: \(b = a \times \frac{\sin B}{\sin A}\).
5. Verify that the calculated side lengths and angles are consistent and possible for a triangle (e.g., all sides positive, angles sum to \$180^\circ$). The sets that satisfy these conditions could form a triangle.