In triangle $qrs$, side $r$ = $83$ cm, angle $\angle r$ = $161°$, and angle $\angle s$ = $5°$. Find the length of side $q$ to the nearest centimeter.

• A. $120$ cm
• B. $78$ cm
• C. $14$ cm
• D. $66$ cm

Answer: (C)

Verified step by step guidance

1. Identify the given elements in triangle QRS: side r = 83 cm, angle at R = 161°, and angle at S = 5°.
2. Calculate the measure of the remaining angle Q using the triangle angle sum property: \(\angle Q = 180^\circ - \angle R - \angle S\).
3. Use the Law of Sines to relate the sides and angles: \(\frac{q}{\sin(\angle R)} = \frac{r}{\sin(\angle Q)}\).
4. Rearrange the Law of Sines formula to solve for side q: \(q = r \times \frac{\sin(\angle R)}{\sin(\angle Q)}\).
5. Substitute the known values of r, \(\angle R\), and \(\angle Q\) into the equation and compute the sine values to find the length of side q.