To find the length of side $q$ in triangle $\Delta QRS$, we can use the Law of Sines, which states:

$\frac{q}{\sin(\angle R)} = \frac{r}{\sin(\angle S)}$

Given:

• $r = 83$ cm
• $\angle R = 161^\circ$
• $\angle S = 5^\circ$

First, we need to find $\angle Q$, the remaining angle in the triangle:

$\angle Q = 180^\circ - \angle R - \angle S = 180^\circ - 161^\circ - 5^\circ = 14^\circ$

Now apply the Law of Sines:

$\frac{q}{\sin(161^\circ)} = \frac{83}{\sin(5^\circ)}$

To find $q$:

$q = \frac{83 \times \sin(161^\circ)}{\sin(5^\circ)}$

Now calculate the sine values and solve for $q$:

$q = \frac{83 \times \sin(161^\circ)}{\sin(5^\circ)} \approx \frac{83 \times 0.1736}{0.0872} \approx \frac{14.4128}{0.0872} \approx 165.2 \text{ cm}$

To the nearest centimeter:

$q \approx 165 \text{ cm}$

The length of side $q$ is approximately 165 cm.

Would you like more details or have any questions?

Here are 5 related questions you might consider:

1. How would the Law of Cosines apply if you had all three sides of the triangle?
2. What happens to the sine values as angles approach 0° or 180°?
3. How would the triangle change if $\angle R$ were slightly less than 161°?
4. What is the significance of knowing the angle sum property of a triangle?
5. How can the Law of Sines be used in non-right triangles?

Tip: When using the Law of Sines, always ensure the triangle's angles are consistent with the triangle inequality theorem.