To determine the number of solutions for the SSA triangle and solve it:

Step 1: Determine if there are 0, 1, or 2 solutions

Using the given data:

• $A = 36^\circ$,
• $a = 23$,
• $b = 16$.

First, calculate the height ($h$) of the triangle: $h = b \cdot \sin(A) = 16 \cdot \sin(36^\circ).$ Then compare $h$, $b$, and $a$ to determine the number of solutions:

1. If $a < h$: No solution.
2. If $a = h$: One right triangle solution.
3. If $h < a < b$: Two solutions.
4. If $a \geq b$: One solution.

Step 2: Calculate $h$

$h = 16 \cdot \sin(36^\circ) \approx 16 \cdot 0.5878 = 9.4048.$ Since $h < a$ and $a > b$, there is only one solution.

Step 3: Solve the triangle

Use the Law of Sines:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}.$ Substitute known values: $\frac{23}{\sin(36^\circ)} = \frac{16}{\sin(B)}.$ Solve for $\sin(B)$: $\sin(B) = \frac{16 \cdot \sin(36^\circ)}{23} \approx \frac{16 \cdot 0.5878}{23} \approx 0.4088.$ Find $B$: $B = \arcsin(0.4088) \approx 24.1^\circ.$

Find $C$:

$C = 180^\circ - A - B = 180^\circ - 36^\circ - 24.1^\circ = 119.9^\circ.$

Find $c$ using the Law of Sines:

$\frac{c}{\sin(C)} = \frac{a}{\sin(A)}.$ Substitute values: $c = \frac{a \cdot \sin(C)}{\sin(A)} = \frac{23 \cdot \sin(119.9^\circ)}{\sin(36^\circ)}.$ $\sin(119.9^\circ) \approx 0.8660, \quad \sin(36^\circ) \approx 0.5878.$ $c \approx \frac{23 \cdot 0.8660}{0.5878} \approx 33.9.$

Final Solution:

• $B = 24.1^\circ$,
• $C = 119.9^\circ$,
• $c = 33.9$.

Answer: B. There is only one possible solution for the triangle: $B = 24.1^\circ, \, C = 119.9^\circ, \, c = 33.9.$

Let me know if you want further details or have additional questions!

Here are 5 related questions to expand on this topic:

1. How do you determine the number of solutions for an SSA triangle?
2. What happens if $a < h$ in an SSA triangle?
3. How is the ambiguous case for SSA triangles resolved?
4. How can the Law of Sines be applied to solve a triangle with given measurements?
5. What are the limitations of the SSA triangle configuration in geometry?

Tip: Always check for the ambiguous case when solving SSA triangles to ensure no solutions are overlooked.