The sides of a hexagon are produced in order. If the measures of exterior angles so obtained are `(6x -1)^@, (10x + 2)^@, (8x+ 2)^@, (9x -3)^@, (5x +4)^@ and (12x +6)^@,` find each exterior angle.

Answer

To solve the problem, we will follow these steps:

### Step 1: Write down the equation for the sum of the exterior angles.
The sum of the exterior angles of any polygon is always equal to 360 degrees. For this hexagon, we have six exterior angles given as:
- \( (6x - 1)^\circ \)
- \( (10x + 2)^\circ \)
- \( (8x + 2)^\circ \)
- \( (9x - 3)^\circ \)
- \( (5x + 4)^\circ \)
- \( (12x + 6)^\circ \)

So, we can set up the equation:
\[ (6x - 1) + (10x + 2) + (8x + 2) + (9x - 3) + (5x + 4) + (12x + 6) = 360 \]

### Step 2: Combine like terms.
Now, let's combine the \(x\) terms and the constant terms:
- \(6x + 10x + 8x + 9x + 5x + 12x = 50x\)
- \(-1 + 2 + 2 - 3 + 4 + 6 = 10\)

Thus, we have:
\[ 50x + 10 = 360 \]

### Step 3: Solve for \(x\).
Now, we will isolate \(x\) by subtracting 10 from both sides:
\[ 50x = 360 - 10 \]
\[ 50x = 350 \]

Next, divide both sides by 50:
\[ x = \frac{350}{50} = 7 \]

### Step 4: Substitute \(x\) back into the expressions for each exterior angle.
Now that we have \(x = 7\), we can find the measure of each exterior angle:

1. For \( (6x - 1)^\circ \):
\[ 6(7) - 1 = 42 - 1 = 41^\circ \]

2. For \( (10x + 2)^\circ \):
\[ 10(7) + 2 = 70 + 2 = 72^\circ \]

3. For \( (8x + 2)^\circ \):
\[ 8(7) + 2 = 56 + 2 = 58^\circ \]

4. For \( (9x - 3)^\circ \):
\[ 9(7) - 3 = 63 - 3 = 60^\circ \]

5. For \( (5x + 4)^\circ \):
\[ 5(7) + 4 = 35 + 4 = 39^\circ \]

6. For \( (12x + 6)^\circ \):
\[ 12(7) + 6 = 84 + 6 = 90^\circ \]

### Final Result:
The measures of the exterior angles are:
- \(41^\circ\)
- \(72^\circ\)
- \(58^\circ\)
- \(60^\circ\)
- \(39^\circ\)
- \(90^\circ\)