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We are tasked with finding side \( a \) of a triangle given the following:

- \( \angle C = 13^\circ \),
- \( \angle A = 22^\circ \),
- Side \( c = 9 \).

### Solution:

1. **Find the third angle \( \angle B \):**
   \[
   \angle B = 180^\circ - \angle A - \angle C
   \]
   \[
   \angle B = 180^\circ - 22^\circ - 13^\circ = 145^\circ
   \]

2. **Apply the Law of Sines:**
   The Law of Sines states:
   \[
   \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
   \]
   Using \( \frac{c}{\sin(C)} \), we solve for \( a \):
   \[
   a = c \cdot \frac{\sin(A)}{\sin(C)}
   \]

3. **Substitute the known values:**
   \[
   a = 9 \cdot \frac{\sin(22^\circ)}{\sin(13^\circ)}
   \]

4. **Calculate the sines and solve:**
   Using a calculator:
   \[
   \sin(22^\circ) \approx 0.3746, \quad \sin(13^\circ) \approx 0.2249
   \]
   \[
   a = 9 \cdot \frac{0.3746}{0.2249} \approx 9 \cdot 1.665 \approx 14.985
   \]

5. **Round to the nearest whole number:**
   \[
   a \approx 15
   \]

### Final Answer:
The side \( a \) is approximately **15**.

Would you like a detailed explanation of any step or further questions?  

Here are five related questions to consider:
1. How is the Law of Sines used in non-right triangles?
2. Why do we need to find the missing angle in a triangle before applying trigonometric laws?
3. What other methods can be used to solve triangles (e.g., Law of Cosines)?
4. Can the Law of Sines determine whether a triangle is obtuse, acute, or right?
5. What happens when given a side and two angles (SAA) in a triangle?

**Tip:** Always ensure your calculator is set to degrees when solving angle-related problems.

For the triangle, m∠C = 13°, m∠A = 22°, c = 9, find side a to the nearest whole number.

Learn how to find side a in a triangle where m∠C = 13°, m∠A = 22°, and side c = 9 using the Law of Sines. This trigonometry problem involves calculating missing angles and applying trigonometric functions, suitable for high school students in Grades 10-12.

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