Answer:
5
Step-by-step explanation:
You want to know the number of sides of a regular polygon that has an exterior angle that is 2/3 of an interior angle.
Let n represent the number of sides of the polygon. Then the exterior angle is 360°/n, and the adjacent interior angle is (180° -360°/n). The given ratio tells us ...
[tex]\dfrac{\text{exterior angle}}{\text{interior angle}}=\dfrac{\dfrac{360}{n}}{180-\dfrac{360}{n}}=\dfrac{2}{3}\\\\\\\dfrac{360}{180n-360}=\dfrac{2}{3}\qquad\text{simplify}\\\\\\\dfrac{2}{n-2}=\dfrac{2}{3}\qquad\text{reduce the fraction}\\\\\\n-2=3\quad\Longrightarrow\quad \boxed{n=5}\qquad\text{match denominators}[/tex]
The polygon has 5 sides.
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Additional comment
We could have solved for the measures of the angles. That would involve the extra step of figuring the number of sides from the angle measure. It seems easier to solve for the answer to the question directly.
The last step "match denominators" is equivalent to multiplying both sides of the equation by (3/2)(n -2).
The exterior and interior angles are 72° and 108°, respectively.
