In a triangle, the sum of the interior angles is always $180^\circ$. Therefore, for triangle $\Delta BCD$, we can write the following equation for the sum of the angles:

$m\angle B + m\angle C + m\angle D = 180^\circ$

Substitute the given expressions for the angles:

$(8x + 13) + (2x + 13) + (x + 11) = 180$

Now, combine like terms:

$8x + 2x + x + 13 + 13 + 11 = 180$ $11x + 37 = 180$

Subtract 37 from both sides:

$11x = 143$

Now, divide by 11:

$x = \frac{143}{11} = 13$

Thus, the value of $x$ is:

$x = 13$

Would you like a breakdown of the steps, or do you have any other questions?

Related Questions:

1. What are the measures of angles $\angle B$, $\angle C$, and $\angle D$ after finding $x$?
2. How would the solution change if one of the angles was given as an exterior angle?
3. Can the same approach be used for non-triangular polygons? Why or why not?
4. How do you solve similar problems for an isosceles or equilateral triangle?
5. What happens if one of the angles was expressed as a quadratic function of $x$?

Tip:

Always remember that the sum of the interior angles of any triangle is $180^\circ$, which is a fundamental rule in geometry.