Geometry

The interior angle must  =120   and the exterior  = 60

So

180 [ n - 2]  / n = 120

180n -360 = 120n

180 n - 120n = 360

60n - 360

n = 360 / 60  =   6  sides

We are given that in a regular polygon, the measure of each interior angle is twice the measure of each exterior angle, and we are asked to find the number of sides of the polygon.

Step 1: Define the variables

Let the number of sides of the polygon be nn. The interior angle θint\theta_{\text{int}} and exterior angle θext\theta_{\text{ext}} of a regular polygon with nn sides are related by the following formulas:

The exterior angle is given by:

θext=360∘n\theta_{\text{ext}} = \frac{360^\circ}{n}

The interior angle is related to the exterior angle by:

θint=180∘−θext\theta_{\text{int}} = 180^\circ - \theta_{\text{ext}}

Step 2: Express the relationship between the angles

We are told that the interior angle is twice the exterior angle. Therefore, we have the equation:

θint=2θext\theta_{\text{int}} = 2 \theta_{\text{ext}}

Substitute θint=180∘−θext\theta_{\text{int}} = 180^\circ - \theta_{\text{ext}} into this equation:

180∘−θext=2θext180^\circ - \theta_{\text{ext}} = 2 \theta_{\text{ext}}

Step 3: Solve for θext\theta_{\text{ext}}

Simplify the equation:

180∘=3θext180^\circ = 3 \theta_{\text{ext}} θext=180∘3=60∘\theta_{\text{ext}} = \frac{180^\circ}{3} = 60^\circ

Step 4: Find the number of sides nn

Now that we know θext=60∘\theta_{\text{ext}} = 60^\circ, we can use the formula for the exterior angle:

θext=360∘n\theta_{\text{ext}} = \frac{360^\circ}{n}

Substitute θext=60∘\theta_{\text{ext}} = 60^\circ into this equation:

60∘=360∘n60^\circ = \frac{360^\circ}{n}

Solve for nn:

n=360∘60∘=6n = \frac{360^\circ}{60^\circ} = 6

Step 5: Conclusion

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The polygon has 6\boxed{6} sides. This is a regular hexagon.