To solve for the length of $AB$ in the triangle, we can use the given information about angles and side lengths.

The triangle has:

• $\angle A = 50^\circ$
• $\angle C = 50^\circ$
• $AB = 4x$
• $BC = 2x + 6$

Since the angles at $A$ and $C$ are both $50^\circ$, the triangle is isosceles. This means that the sides opposite those angles must be equal. Therefore, $AB = BC$.

Setting up the equation:

$4x = 2x + 6$

Now, solve for $x$:

1. Subtract $2x$ from both sides:

$4x - 2x = 6$ $2x = 6$

2. Divide both sides by 2:

$x = 3$

Finding the length of $AB$:

Since $AB = 4x$, substitute $x = 3$ into this expression:

$AB = 4(3) = 12$

Thus, the length of $AB$ is 12. The correct answer is C. 12.

Would you like a more detailed explanation of the isosceles triangle property or any other aspect of this problem?

Here are five related questions you might explore:

1. What is the sum of the angles in any triangle, and why?
2. How would you find the perimeter of this triangle?
3. What would happen if $\angle A$ and $\angle C$ were not equal?
4. How can you apply the Law of Sines to solve triangles with non-right angles?
5. If you knew the area of the triangle, how would you find the height?

Tip: When solving for unknowns in isosceles triangles, always use the property that two sides (opposite equal angles) are congruent—it simplifies the equation.