How Do You Find The Exterior Angle

How Do You Find the Exterior Angle? A Comprehensive Guide

Understanding exterior angles is crucial in geometry, offering a pathway to solve complex problems involving polygons and their properties. This comprehensive guide will delve deep into the concept of exterior angles, explaining how to find them in various shapes, and exploring their relationship with interior angles. We'll cover different approaches, providing you with the tools to confidently tackle any exterior angle problem.

What is an Exterior Angle?

An exterior angle is formed when one side of a polygon is extended. It's the angle created outside the polygon between the extended side and the adjacent side. Each vertex of a polygon has two exterior angles, one on either side of the extended line. However, we typically focus on one exterior angle at a time, usually the one that creates a linear pair with the interior angle at that vertex.

A linear pair refers to two angles that are adjacent and supplementary, meaning their measures add up to 180 degrees. This relationship is fundamental to understanding how exterior and interior angles are connected.

Finding Exterior Angles: Methods and Examples

The method for finding an exterior angle depends on the information given. Here are the key approaches:

1. Using the Relationship with the Interior Angle

This is the most common and straightforward method. Since an exterior angle and its corresponding interior angle form a linear pair, their sum is always 180 degrees. Therefore:

Exterior Angle = 180° - Interior Angle

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

2. Using the Sum of Exterior Angles

The sum of the exterior angles of any polygon, one at each vertex, always equals 360 degrees. This fact is incredibly useful when dealing with regular polygons (polygons with all sides and angles equal) or when you know the measure of several exterior angles.

Sum of Exterior Angles = 360°

Example: A hexagon has six exterior angles. If five of them measure 50°, 60°, 70°, 80°, and 90°, the sixth exterior angle can be found by subtracting the sum of the known angles from 360°: 360° - (50° + 60° + 70° + 80° + 90°) = 110°.

3. Using the Formula for Regular Polygons

For regular polygons, where all sides and angles are equal, calculating the exterior angle is particularly simple:

Exterior Angle = 360° / Number of Sides

Example: A regular pentagon has 5 sides. Its exterior angle is 360° / 5 = 72°. This means each interior angle of a regular pentagon measures 180° - 72° = 108°.

4. Using the Properties of Triangles

Exterior angles are closely linked to the properties of triangles. Consider a triangle with one side extended. The exterior angle formed is equal to the sum of the two opposite interior angles. This is known as the Exterior Angle Theorem.

Exterior Angle = Sum of Opposite Interior Angles

Example: In a triangle with interior angles of 60°, 70°, and 50°, the exterior angle at the vertex with the 50° interior angle is 60° + 70° = 130°.

This theorem can be extended to more complex polygons by dividing them into triangles.

Tackling Complex Scenarios: Irregular Polygons and Advanced Techniques

Dealing with irregular polygons requires a more nuanced approach. You might need to utilize a combination of the methods discussed above or employ more advanced geometric principles.

For instance, if you're given some interior angles of an irregular polygon but not all of them, you can use the property that the sum of the interior angles of an n-sided polygon is (n-2) * 180°. This allows you to find the missing interior angles and then calculate the corresponding exterior angles.

Let's consider a more complex example:

Example: An irregular quadrilateral has interior angles of 80°, 100°, and 120°. Find the measure of the fourth interior angle and all exterior angles.

1. Find the fourth interior angle: The sum of interior angles in a quadrilateral (4 sides) is (4-2) * 180° = 360°. Therefore, the fourth interior angle is 360° - (80° + 100° + 120°) = 60°.

2. Find the exterior angles: Using the linear pair relationship:

   • Exterior angle adjacent to 80°: 180° - 80° = 100°
   • Exterior angle adjacent to 100°: 180° - 100° = 80°
   • Exterior angle adjacent to 120°: 180° - 120° = 60°
   • Exterior angle adjacent to 60°: 180° - 60° = 120°

Notice that the sum of these exterior angles is 360°, confirming the general property.

Applications of Exterior Angles

Understanding exterior angles isn't just an abstract mathematical concept; it has practical applications across various fields:

• Surveying and Navigation: Exterior angles are crucial in surveying to accurately measure land and create maps. They are also used in navigation to determine bearings and directions.

• Architecture and Engineering: Exterior angles play a role in structural design, ensuring the stability and strength of buildings and other constructions.

• Computer Graphics and Game Development: Exterior angles are used extensively in computer graphics and game development to create realistic 3D models and simulations. They help determine the orientation and positioning of objects in a virtual environment.

• Crystallography: In crystallography, exterior angles help determine the angles between crystal faces and the overall symmetry of a crystal structure.

Troubleshooting Common Mistakes

• Confusing Interior and Exterior Angles: Ensure you are working with the correct angle. Remember, exterior angles are formed outside the polygon, while interior angles are inside.

• Incorrectly Applying the Sum of Exterior Angles: Remember this sum is always 360°, regardless of the number of sides of the polygon.

• Forgetting Linear Pair Property: Don't forget that an interior and its corresponding exterior angle sum to 180°.

• Assuming Irregular Polygons are Regular: Irregular polygons have unequal sides and angles; therefore, you can't use the simple formula for regular polygons.

Conclusion: Mastering Exterior Angles

Finding exterior angles is a fundamental skill in geometry, applicable in numerous contexts. By understanding the relationship between exterior and interior angles, the sum of exterior angles, and the properties of regular and irregular polygons, you can confidently solve various problems involving these crucial geometric concepts. Remember to practice consistently, and you'll soon master the art of calculating exterior angles. The more you practice, the more intuitive this will become, enabling you to approach even complex geometric problems with confidence and accuracy.