To solve for [tex]\( x \)[/tex] given that each exterior angle of a regular decagon has a measure of [tex]\( (3x + 6)^\circ \)[/tex], we need to understand a few key properties of regular polygons.
1. Exterior Angle of a Regular Polygon:
The exterior angle of a regular polygon with [tex]\( n \)[/tex] sides can be calculated using the formula:
[tex]\[
\text{exterior angle} = \frac{360^\circ}{n}
\][/tex]
2. Applying to a Decagon:
For a regular decagon, which has [tex]\( 10 \)[/tex] sides, the measure of each exterior angle is:
[tex]\[
\text{exterior angle} = \frac{360^\circ}{10} = 36^\circ
\][/tex]
3. Setting Up the Equation:
We are given that the exterior angle is [tex]\( (3x + 6)^\circ \)[/tex]. Therefore, we can set up the following equation:
[tex]\[
3x + 6 = 36
\][/tex]
4. Solving for [tex]\( x \)[/tex]:
To find the value of [tex]\( x \)[/tex], we solve the equation [tex]\( 3x + 6 = 36 \)[/tex]:
Subtract 6 from both sides:
[tex]\[
3x = 30
\][/tex]
Divide both sides by 3:
[tex]\[
x = 10
\][/tex]
5. Conclusion:
Therefore, the value of [tex]\( x \)[/tex] that satisfies the given conditions is:
[tex]\[
x = 10
\][/tex]
So the correct option is [tex]\( x = 10 \)[/tex].
