To find the size of angle $A$, we can use the sine rule, since we know two sides and the angle opposite one of them. The sine rule is given by:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Here:

• $a = 9 \, \text{cm}$ (side opposite angle $B = 38^\circ$),
• $b = 11 \, \text{cm}$,
• and we need to find angle $A$.

We can first use the sine rule to find angle $C$. The sine rule for angle $C$ is:

$\frac{c}{\sin C} = \frac{a}{\sin B}$

$\frac{11}{\sin C} = \frac{9}{\sin 38^\circ}$

Let me calculate $\sin C$ and then proceed to find angle $A$.The measure of angle $C$ is approximately $48.81^\circ$.

Now, to find angle $A$, we can use the fact that the sum of angles in a triangle is always $180^\circ$. Thus:

$A = 180^\circ - B - C$

Let me calculate angle $A$.The size of angle $A$ is approximately $93.19^\circ$.

Would you like more details on the process or have any other questions? Here are some related questions to explore:

1. How can the sine rule be applied to other triangles?
2. What is the cosine rule, and when is it used?
3. How do you solve for angles in non-right triangles?
4. What are the different types of triangles, and how do they affect angle calculations?
5. How do angle properties in polygons differ from those in triangles?

Tip: Always remember that the angles of any triangle must sum to 180 degrees. This can help double-check your work!