Given triangle $VUW$ with side $v$ opposite angle $V$, side $u$ opposite angle $U$, and side $w$ opposite angle $W$, if $v=7$, $u=10$, and $W=30°$, what are the measures of angles $V$ and $U$ (rounded to the nearest degree)?

• A. $V=44°,U=106°$
• B. $V=60°,U=90°$
• C. $V=30°,U=120°$
• D. $V=53°,U=97°$

Answer: (A)

Verified step by step guidance

1. Identify the known elements: side \(v = 7\) opposite angle \(V\), side \(u = 10\) opposite angle \(U\), and angle \(W = 30^\circ\) opposite side \(w\) (unknown).
2. Use the Law of Sines, which states \(\frac{v}{\sin V} = \frac{u}{\sin U} = \frac{w}{\sin W}\), to relate the sides and angles of the triangle.
3. Express \(\sin V\) and \(\sin U\) in terms of \(w\) and known values: \(\sin V = \frac{v}{w} \sin W\) and \(\sin U = \frac{u}{w} \sin W\).
4. Apply the triangle angle sum property: \(V + U + W = 180^\circ\), so \(V + U = 180^\circ - 30^\circ = 150^\circ\).
5. Use the relationships from the Law of Sines and the angle sum to solve for angles \(V\) and \(U\), typically by first finding \(w\) using the Law of Cosines or by iterative/trigonometric methods, then calculating \(V\) and \(U\) using inverse sine functions and rounding to the nearest degree.