The exterior angles of a convex pentagon measures (18x +12°), 16X degrees, (8x+6°), (10x - 12°) and (5X + 12°) determine the measure of the largest interior angle

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.

Learn how to calculate the largest interior angle of a convex pentagon using given exterior angles. Detailed step-by-step solution provided, covering the formulas for the sum of exterior angles and interior angles of a polygon. Suitable for high school students and those interested in geometry.

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