The problem involves an isosceles triangle $\triangle ABC$, where angles $\angle A$ and $\angle C$ are given as $65^\circ$, and sides $AB = 6x$ and $BC = 3x + 9$. We are asked to find the length of side $AB$.

Step-by-step solution:

1. Since the triangle is isosceles, sides $AB$ and $BC$ are equal. Therefore, we can set the expressions for these sides equal: $6x = 3x + 9$

2. Solve for $x$: $6x - 3x = 9$ $3x = 9$ $x = 3$

3. Substitute $x = 3$ back into the expression for $AB$ to find the length of $AB$: $AB = 6x = 6(3) = 18$

Thus, the length of side $AB$ is $18$.

The correct answer is C. 18.

Do you have any questions, or would you like more details on any of the steps?

Here are five related questions to expand on this topic:

1. How can we determine if a triangle is isosceles using only its angles?
2. What is the sum of the internal angles of any triangle?
3. How do you solve for unknown variables in other types of triangles (e.g., scalene, equilateral)?
4. How can the Pythagorean theorem be applied in right triangles?
5. What are the properties of equilateral triangles, and how do they differ from isosceles triangles?

Tip: Always look for symmetry in problems involving isosceles triangles, as it helps simplify equations and relationships between sides and angles.