To find the value of [tex]\(x\)[/tex] given that each exterior angle of a regular decagon (10-sided polygon) is [tex]\((3x + 6)^{\circ}\)[/tex], follow these steps:
1. Understanding the Exterior Angles of a Decagon:
- The sum of all exterior angles of any polygon is always [tex]\(360^\circ\)[/tex].
- For a regular polygon, each exterior angle is equal. Since a decagon has 10 sides, each exterior angle for the decagon is given by:
[tex]\[
\text{Each exterior angle} = \frac{360^\circ}{10}
\][/tex]
- Simplifying this, we get:
[tex]\[
\text{Each exterior angle} = 36^\circ
\][/tex]
2. Setting Up the Equation:
- Given that each exterior angle of the regular decagon is [tex]\((3x + 6)^\circ\)[/tex]:
[tex]\[
3x + 6 = 36
\][/tex]
3. Solving for [tex]\(x\)[/tex]:
- Subtract 6 from both sides of the equation:
[tex]\[
3x = 36 - 6
\][/tex]
- Simplifying the right-hand side:
[tex]\[
3x = 30
\][/tex]
- Finally, divide both sides by 3:
[tex]\[
x = \frac{30}{3} = 10
\][/tex]
Thus, the value of [tex]\(x\)[/tex] is [tex]\(10\)[/tex]. This corresponds to the option:
[tex]\[
x = 10
\][/tex]
So, the correct answer is [tex]\( \boxed{10} \)[/tex].
