This problem involves finding the value of $x$ using the angles provided in a polygon. Here's the step-by-step process:

Step 1: Recognize the geometric property

The figure appears to be a quadrilateral. The sum of the interior angles of any quadrilateral is:

$360^\circ$

Step 2: Write an equation for the sum of the angles

The angles given in the figure are:

• $(x + 11)^\circ$
• $(x + 14)^\circ$
• $(x + 5)^\circ$
• $(3x - 9)^\circ$
• $78^\circ$
• $(2x + 13)^\circ$

Adding all these angles together:

$(x + 11) + (x + 14) + (x + 5) + (3x - 9) + 78 + (2x + 13) = 360$

Step 3: Simplify the equation

Combine all the $x$-terms and constants:

$x + x + x + 3x + 2x + 11 + 14 + 5 - 9 + 78 + 13 = 360$

$8x + 112 = 360$

Step 4: Solve for $x$

Subtract 112 from both sides:

$8x = 248$

Divide by 8:

$x = 31$

Final Answer:

The value of $x$ is:

$x = 31$

Would you like further details or explanations?

Here are 5 related questions to expand your understanding:

1. How do you calculate the sum of the interior angles of a polygon?
2. What happens if the given figure is not a quadrilateral? How would you solve it?
3. How can you verify your solution for $x$?
4. What role does algebra play in solving geometric problems like this?
5. Can you solve for the individual angle measures with the found value of $x$?

Tip: Always double-check the total sum of the angles when solving polygon problems!