How Do You Find The Exterior Angle

How Do You Find the Exterior Angle? A Comprehensive Guide

Understanding exterior angles is crucial in geometry, particularly when dealing with polygons. This comprehensive guide will delve into the various methods for finding exterior angles, catering to different levels of understanding, from basic concepts to more advanced applications. We'll cover different polygon types, explore the relationship between interior and exterior angles, and provide numerous examples to solidify your understanding.

Understanding Interior and Exterior Angles

Before we jump into calculating exterior angles, let's clarify the terminology. An interior angle is an angle inside a polygon, formed by two adjacent sides. An exterior angle, on the other hand, is formed by one side of a polygon and the extension of an adjacent side. It lies outside the polygon. Crucially, an interior angle and its corresponding exterior angle are supplementary – meaning they add up to 180 degrees.

Visualizing Interior and Exterior Angles

Imagine a triangle. Each vertex has an interior angle. If you extend one side of the triangle at a vertex, the angle formed outside the triangle is the exterior angle for that vertex. This relationship holds true for all polygons, not just triangles.

Methods for Finding Exterior Angles

There are several approaches to calculating exterior angles, depending on the information provided.

1. Using the Relationship with Interior Angles

This is the most fundamental method. Since an interior angle and its corresponding exterior angle are supplementary, we can use the following formula:

Exterior Angle = 180° - Interior Angle

Example: If an interior angle of a polygon measures 120°, its corresponding exterior angle is 180° - 120° = 60°.

This method is particularly useful when you already know the measure of the interior angle.

2. Using the Sum of Exterior Angles

A remarkable property of polygons is that the sum of their exterior angles (one at each vertex) always equals 360°. This holds true regardless of the number of sides the polygon has. This fact provides another pathway to find individual exterior angles.

Example: Consider a regular pentagon (5 sides). Since the sum of its exterior angles is 360°, each exterior angle (because it's regular, all angles are equal) is 360°/5 = 72°.

This method is especially efficient for regular polygons (polygons with all sides and angles equal).

3. Using the Number of Sides (for Regular Polygons Only)

For regular polygons, we can directly calculate the exterior angle using the number of sides (n):

Exterior Angle = 360° / n

This formula is a direct consequence of the sum of exterior angles being 360°.

Example: A regular hexagon (n=6) has an exterior angle of 360°/6 = 60°.

This is a concise method, but remember, it only applies to regular polygons.

4. Using Geometry Theorems (for Specific Polygons)

Depending on the type of polygon, specific geometric theorems can help determine exterior angles. For example, in an isosceles triangle, if you know one base angle, you can easily find the other base angle and subsequently the exterior angle. Similar approaches exist for other types of triangles and quadrilaterals.

Example: In an isosceles triangle with base angles of 70°, the third angle is 180° - (70° + 70°) = 40°. The exterior angle at the vertex with the 40° angle is 180° - 40° = 140°.

This method requires understanding the specific properties of the polygon in question.

Advanced Applications and Problem Solving

The methods described above form the bedrock for solving more complex problems involving exterior angles.

1. Irregular Polygons

For irregular polygons (polygons where sides and angles are not equal), the sum of exterior angles remains 360°. However, you cannot directly use the formula 360°/n to find individual exterior angles. Instead, you'll need to find the interior angles first (using other geometric principles or provided information) and then use the supplementary angle relationship (180° - Interior Angle) to calculate the exterior angles.

2. Composite Figures

When dealing with composite figures (shapes formed by combining several polygons), break down the figure into its constituent polygons. Find the exterior angles of each individual polygon, and then use these to determine the overall exterior angles of the composite figure as needed. This requires careful visualization and application of angle relationships.

3. Proofs and Deductive Reasoning

Exterior angles are often used in geometric proofs. Understanding the relationships between interior and exterior angles, and the consistent 360° sum of exterior angles, is essential for constructing logical arguments and solving geometrical problems.

Practical Examples

Let's solidify our understanding with some practical examples.

Example 1: A regular octagon has how many sides? What is the measure of each exterior angle?

• Solution: A regular octagon has 8 sides (n=8). The measure of each exterior angle is 360°/8 = 45°.

Example 2: One interior angle of a pentagon measures 110°. What is the measure of its corresponding exterior angle?

• Solution: The exterior angle is 180° - 110° = 70°.

Example 3: A triangle has interior angles of 60°, 70°, and 50°. What are the measures of its exterior angles?

• Solution: The exterior angles are: 180° - 60° = 120°, 180° - 70° = 110°, and 180° - 50° = 130°. Notice that 120° + 110° + 130° = 360°.

Example 4: A quadrilateral has exterior angles of 75°, 85°, and 100°. What is the measure of the fourth exterior angle?

• Solution: The sum of exterior angles is 360°. Therefore, the fourth exterior angle is 360° - (75° + 85° + 100°) = 100°.

Conclusion

Finding exterior angles is a fundamental skill in geometry. By mastering the methods outlined in this guide, you'll be equipped to tackle various problems involving polygons, regardless of their regularity or complexity. Remember the key relationships: the supplementary relationship between interior and exterior angles, and the constant 360° sum of exterior angles. Practice regularly with diverse examples to solidify your understanding and build confidence in your ability to solve geometric problems efficiently and accurately. This knowledge forms a strong foundation for more advanced geometric concepts and problem-solving.