Let's denote the number of sides of the polygon as $n$.

Step 1: Relationship between interior and exterior angles

• The exterior angle of any polygon is given by the formula: $\text{Exterior angle} = \frac{360^\circ}{n}$
• The interior angle of a polygon is related to the exterior angle by: $\text{Interior angle} = 180^\circ - \text{Exterior angle}$

Step 2: Set up the equation

According to the problem, the measure of each interior angle is four times the measure of each exterior angle. Therefore, we can set up the following equation: $\text{Interior angle} = 4 \times \text{Exterior angle}$ Substitute the expressions for the interior and exterior angles: $180^\circ - \frac{360^\circ}{n} = 4 \times \frac{360^\circ}{n}$

Step 3: Solve for $n$

Now, let's solve the equation: $180^\circ - \frac{360^\circ}{n} = \frac{1440^\circ}{n}$ Multiply both sides by $n$ to eliminate the denominators: $180n - 360 = 1440$ Add 360 to both sides: $180n = 1800$ Now, divide both sides by 180: $n = 10$

Final Answer:

The polygon has 10 sides.

Would you like further details on the steps or have any questions?

---

Here are some related questions for you:

1. How would you calculate the interior and exterior angles of a polygon with a known number of sides?
2. What happens to the interior and exterior angles as the number of sides increases?
3. Can you derive the formula for the sum of the interior angles of any polygon?
4. How does the relationship between interior and exterior angles change in polygons with different numbers of sides?
5. How do you prove that the sum of all exterior angles of any polygon is always 360°?

Tip: For polygons where interior and exterior angles are related by a factor, try using the angle relationships and simplify the equation step by step.