To find the value of [tex]\( x \)[/tex] when given that each exterior angle of a regular decagon (a 10-sided polygon) is [tex]\((3x + 6)^{\circ}\)[/tex], follow these steps:
1. Determine the measure of each exterior angle in a regular decagon:
A regular decagon has [tex]\( 10 \)[/tex] sides. The sum of all exterior angles of any polygon is always [tex]\( 360^{\circ} \)[/tex]. Therefore, each exterior angle of a regular decagon is given by:
[tex]\[
\text{Each exterior angle} = \frac{360^{\circ}}{10} = 36^{\circ}
\][/tex]
2. Set up the equation:
We are given that each exterior angle has the measure [tex]\( (3x + 6)^{\circ} \)[/tex]. So we set up the equation:
[tex]\[
3x + 6 = 36
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Subtract 6 from both sides of the equation:
[tex]\[
3x = 30
\][/tex]
Divide both sides by 3:
[tex]\[
x = 10
\][/tex]
4. Verify the solution:
Substitute [tex]\( x = 10 \)[/tex] back into the expression for the angle to check:
[tex]\[
3(10) + 6 = 30 + 6 = 36
\][/tex]
This matches our calculated measure for each exterior angle of the decagon.
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{10} \)[/tex].
