To solve this problem, let's start by using the relationship between the interior and exterior angles of a regular polygon.

Step 1: Define the Variables

• Let the measure of the exterior angle of the polygon be $x$ degrees.
• Then, the measure of the interior angle will be $2x$ degrees (since it is given that the interior angle is twice the exterior angle).

Step 2: Use the Relationship Between Interior and Exterior Angles

• The sum of the interior angle and the exterior angle of any polygon is always $180^\circ$.

  Therefore, we have: $2x + x = 180^\circ$

Step 3: Solve for $x$

$3x = 180^\circ$ $x = \frac{180^\circ}{3} = 60^\circ$

So, the exterior angle $x$ is $60^\circ$.

Step 4: Find the Interior Angle

$\text{Interior Angle} = 2x = 2 \times 60^\circ = 120^\circ$

Step 5: Determine the Name of the Polygon

• The exterior angle of a regular polygon is given by: $\text{Exterior Angle} = \frac{360^\circ}{n}$ where $n$ is the number of sides.

Given that the exterior angle is $60^\circ$: $\frac{360^\circ}{n} = 60^\circ$ $n = \frac{360^\circ}{60^\circ} = 6$

So, the polygon is a regular hexagon.

Final Answers:

a. The measure of the interior angle is 120°. b. The measure of the exterior angle is 60°. c. The polygon is a regular hexagon.

Would you like further details or explanations?

Here are 5 related questions to practice:

1. What is the measure of each interior angle in a regular octagon?
2. How many sides does a regular polygon have if each exterior angle measures 45°?
3. What is the sum of the interior angles of a regular dodecagon (12-sided polygon)?
4. If an exterior angle of a regular polygon is 24°, what is the name of the polygon?
5. How would the problem change if the interior angle was three times the exterior angle?

Tip: Remember that the sum of the exterior angles of any polygon is always 360°, regardless of the number of sides!