To solve for $a$ in this triangle, we can use similar triangles, as the two triangles formed are proportional based on the given dimensions.

1. Identify the proportions: Since the two triangles share a common angle and are similar, the sides are proportional. Let’s set up the proportion using the sides of the triangles.

   • The sides corresponding to $a$ and 12.6 are 6.1 and 9.6, respectively.

2. Set up the proportion:

   $\frac{a}{12.6} = \frac{6.1}{9.6}$

3. Solve for $a$:

   $a = 12.6 \times \frac{6.1}{9.6}$

4. Calculate:

   $a = 12.6 \times 0.6354 \approx 8.0$

Thus, $a \approx 8.0$ (rounded to 1 decimal place).

Would you like further details or have any questions about the solution process?

Here are five related questions that expand upon this concept:

1. How do you determine if two triangles are similar?
2. Can you find the missing side of a triangle if only one angle and one side are known?
3. How would the solution change if the length of $a$ was given and you needed to find another side?
4. What are some common real-life applications of similar triangles?
5. How would this problem be solved if the angle measures were provided instead of side lengths?

Tip: Always check for similar triangles in geometric problems, as they can greatly simplify finding missing lengths.