Question

1. the sum of the measures of the interior angles of a polygon is twice the measure of the sum of it’s exterior angles. what is the name of this polygon?

Explanation:

Step1: Recall exterior angle sum

The sum of exterior angles of any polygon is \(360^\circ\).

Step2: Set up interior angle sum equation

Let the sum of interior angles be \(S\). Given \(S = 2\times360^\circ=720^\circ\).

Step3: Use interior angle formula

The formula for the sum of interior angles of a polygon with \(n\) sides is \(S=(n - 2)\times180^\circ\). Substitute \(S = 720^\circ\):
\((n - 2)\times180^\circ=720^\circ\)

Step4: Solve for \(n\)

Divide both sides by \(180^\circ\): \(n - 2=\frac{720^\circ}{180^\circ}=4\)
Add 2 to both sides: \(n = 4 + 2=6\)
A polygon with 6 sides is a hexagon.

Answer:

Hexagon