https://file.aiquickdraw.com/mathbot/file/1726891008732vlbpbfex.jpg

The problem in the image asks to determine whether the following statement is **ALWAYS**, **SOMETIMES**, or **NEVER true**:  
**"The sum of the measures of the interior angles of a polygon is equal to the sum of the measures of its exterior angles."**

### Analysis:

1. **Interior Angles of a Polygon**:  
   The sum of the interior angles of an \(n\)-sided polygon is given by:
   \[
   \text{Sum of interior angles} = 180^\circ \times (n - 2)
   \]
   This formula holds for any polygon with \(n\) sides.

2. **Exterior Angles of a Polygon**:  
   The sum of the exterior angles of any polygon, regardless of the number of sides, is always:
   \[
   \text{Sum of exterior angles} = 360^\circ
   \]
   This is true whether the polygon is regular or irregular.

### Conclusion for the Statement:
The sum of the interior angles changes depending on the number of sides, while the sum of the exterior angles is always 360°. Hence, the sum of interior angles is **not equal** to the sum of exterior angles in most cases.

This means the statement "The sum of the measures of the interior angles of a polygon is equal to the sum of the measures of its exterior angles" is **NEVER true** for polygons with more than 3 sides.

---

#### Proof Outline (Example 4 in the image):
- The task is to prove that the sum of the exterior angles of any polygon is **360°**.
- The diagram shows a polygon with 5 sides (pentagon) and angles labeled.
- For each interior angle (e.g., \(a\), \(b\), etc.), the exterior angle is supplementary (i.e., their sum is \(180^\circ\)) due to the linear pair rule.
- This property generalizes to all polygons.

#### Would you like a detailed explanation of this proof or any further clarifications?

---

### Related Questions:

1. What is the formula for the sum of interior angles for a regular polygon with \(n\) sides?
2. How can you calculate the measure of each exterior angle for a regular polygon?
3. Can the sum of the interior angles ever be less than 360° for any polygon?
4. How does the sum of the exterior angles of a polygon relate to a full circle?
5. In what types of polygons would interior and exterior angles form supplementary pairs?

### Tip:
Always remember that no matter how many sides a polygon has, the sum of its exterior angles is always 360°.

Discuss with a partner if the statement 'The sum of the measures of the interior angles of a polygon is equal to the sum of the measures of its exterior angles' is always, sometimes, or never true. Justify your answer.

This lesson explores whether the sum of the interior angles of a polygon is equal to the sum of its exterior angles. Key geometric concepts such as the interior and exterior angle formulas and the Linear Pair Theorem are discussed. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation