To solve this problem, we'll use the fact that the sum of the exterior angles of any convex polygon is always $360^\circ$. Given the exterior angles of the pentagon, we can set up an equation and solve for $x$. Then, we can find the measures of the exterior angles and subsequently the interior angles. The largest interior angle will be determined from these calculations.

Step-by-Step Solution

1. Sum of the Exterior Angles: The sum of the exterior angles of any convex polygon is $360^\circ$. Therefore, we can write the equation: $(18x + 12^\circ) + 16x^\circ + (8x + 6^\circ) + (10x - 12^\circ) + (5x + 12^\circ) = 360^\circ$

2. Simplify the Equation: Combine like terms: $18x + 12 + 16x + 8x + 6 + 10x - 12 + 5x + 12 = 360$ $(18x + 16x + 8x + 10x + 5x) + (12 + 6 - 12 + 12) = 360$ $57x + 18 = 360$

3. Solve for $x$: Isolate $x$: $57x + 18 = 360$ $57x = 342$ $x = 6$

4. Calculate the Measures of the Exterior Angles: Substitute $x = 6$ into each expression: $18x + 12^\circ = 18(6) + 12 = 108 + 12 = 120^\circ$ $16x^\circ = 16(6) = 96^\circ$ $8x + 6^\circ = 8(6) + 6 = 48 + 6 = 54^\circ$ $10x - 12^\circ = 10(6) - 12 = 60 - 12 = 48^\circ$ $5x + 12^\circ = 5(6) + 12 = 30 + 12 = 42^\circ$

5. Calculate the Interior Angles: The interior angle of a polygon is given by $180^\circ$ minus the exterior angle. $\text{Interior angle corresponding to } 120^\circ = 180^\circ - 120^\circ = 60^\circ$ $\text{Interior angle corresponding to } 96^\circ = 180^\circ - 96^\circ = 84^\circ$ $\text{Interior angle corresponding to } 54^\circ = 180^\circ - 54^\circ = 126^\circ$ $\text{Interior angle corresponding to } 48^\circ = 180^\circ - 48^\circ = 132^\circ$ $\text{Interior angle corresponding to } 42^\circ = 180^\circ - 42^\circ = 138^\circ$

6. Determine the Largest Interior Angle: Among the interior angles calculated: $60^\circ, 84^\circ, 126^\circ, 132^\circ, 138^\circ$ The largest interior angle is $138^\circ$.

Therefore, the measure of the largest interior angle is $138^\circ$.

Would you like further details or have any questions?

Here are some more questions you might find interesting:

1. How do you find the sum of the interior angles of a polygon?
2. What is the relationship between interior and exterior angles of a polygon?
3. How do you calculate the measure of each interior angle of a regular polygon?
4. How do you solve equations with multiple terms involving a variable?
5. How can you verify the solution of an equation?

Tip: When working with angles in polygons, remember the fundamental properties and relationships between interior and exterior angles to simplify your calculations.