MATH135_R6

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The measure of an interior angle of a regular polygon is twice as large as the measure of an exterior angle.
a. What is the measure of the interior angle?
b. What is the measure of the exterior angle?
c. What is the name of the polygon?

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!

Learn how to find the measures of interior and exterior angles of a regular polygon. This problem involves understanding the relationship between interior and exterior angles and identifying the type of polygon based on given angle measures. Suitable for high school students.

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