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A 30-sided regular polygon has an angle measure represented as [tex][tex]$6g^{\circ}$[/tex][/tex]. Determine the value of [tex]g[/tex].
A. [tex]g=30[/tex] B. [tex]g=28[/tex] C. [tex]g=180[/tex] D. [tex]g=168[/tex]
To determine the value of [tex]\( g \)[/tex] for a 30-sided regular polygon where the interior angle is given as [tex]\( 6g \)[/tex] degrees, we can follow these steps:
1. Determine the Measure of an Interior Angle: The formula to find the sum of the interior angles of an [tex]\( n \)[/tex]-sided polygon is: [tex]\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\][/tex] For a polygon with 30 sides ([tex]\( n = 30 \)[/tex]): [tex]\[
\text{Sum of interior angles} = (30 - 2) \times 180^\circ = 28 \times 180^\circ = 5040^\circ
\][/tex] Since it is a regular polygon, all interior angles are equal. Therefore, the measure of one interior angle is: [tex]\[
\text{Interior angle} = \frac{\text{Sum of interior angles}}{n} = \frac{5040^\circ}{30} = 168^\circ
\][/tex]
2. Set Up the Equation: We know the interior angle is represented as [tex]\( 6g \)[/tex] degrees. Thus: [tex]\[
6g = 168^\circ
\][/tex]
3. Solve for [tex]\( g \)[/tex]: Divide both sides of the equation by 6: [tex]\[
g = \frac{168^\circ}{6} = 28
\][/tex]
Thus, the value of [tex]\( g \)[/tex] is [tex]\( \boxed{28} \)[/tex].
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