How Do You Find Exterior Angles

How Do You Find Exterior Angles? A Comprehensive Guide

Exterior angles are a fundamental concept in geometry, playing a crucial role in understanding polygon properties and solving geometric problems. This comprehensive guide will delve into the intricacies of exterior angles, explaining how to find them in various polygon types, providing clear examples, and exploring their applications. We will cover both regular and irregular polygons, ensuring a thorough understanding for all readers.

Understanding Exterior Angles

Before we dive into the methods of finding exterior angles, let's establish a clear definition. An exterior angle is formed by extending one side of a polygon. It's the angle between the extended side and the adjacent side of the polygon. Each vertex of a polygon has two exterior angles, one formed by extending each side. However, we typically focus on one exterior angle at each vertex, which forms a linear pair with the interior angle.

A linear pair consists of two adjacent angles whose non-common sides form a straight line. The sum of angles in a linear pair is always 180°. This relationship between interior and exterior angles is crucial for calculating exterior angles.

Key Properties of Exterior Angles

• Supplementary to Interior Angles: As mentioned, an exterior angle and its adjacent interior angle are supplementary, meaning their sum equals 180°.
• Sum of Exterior Angles: The sum of the exterior angles of any convex polygon (a polygon where all interior angles are less than 180°) is always 360°. This is a fundamental theorem in geometry and holds true regardless of the number of sides the polygon has.
• Relationship to Interior Angles: The measure of an exterior angle is equal to the sum of the measures of the two opposite interior angles. This property is particularly useful in triangles.

Finding Exterior Angles in Different Polygons

Let's explore the methods for finding exterior angles in different polygon types:

1. Triangles

Finding exterior angles in triangles is relatively straightforward. Since the sum of interior angles in a triangle is 180°, and exterior and interior angles are supplementary, we can easily calculate the exterior angles:

Method 1: Using Supplementary Angles

• Step 1: Find the measure of the interior angle adjacent to the exterior angle.
• Step 2: Subtract the interior angle's measure from 180°. The result is the measure of the exterior angle.

Example:

If an interior angle of a triangle measures 70°, its adjacent exterior angle measures 180° - 70° = 110°.

Method 2: Using the Remote Interior Angles

• Step 1: Identify the two interior angles that are not adjacent to the exterior angle (these are called remote interior angles).
• Step 2: Add the measures of the remote interior angles. The sum is equal to the measure of the exterior angle.

Example:

If the remote interior angles of a triangle measure 50° and 60°, the exterior angle measures 50° + 60° = 110°.

2. Quadrilaterals

Quadrilaterals have four sides and four interior angles. The sum of their interior angles is 360°. To find the exterior angles:

Method 1: Using Supplementary Angles

Similar to triangles, subtract each interior angle from 180° to find its corresponding exterior angle.

Method 2: Using the Sum of Exterior Angles

Since the sum of exterior angles of any polygon is 360°, if you know three exterior angles, you can find the fourth by subtracting the sum of the known angles from 360°.

Example:

If three exterior angles of a quadrilateral measure 70°, 80°, and 100°, the fourth exterior angle measures 360° - (70° + 80° + 100°) = 110°.

3. Regular Polygons

Regular polygons have all sides and angles equal in measure. This simplifies the calculation of exterior angles significantly:

Method: Using the Formula

The measure of each exterior angle of a regular polygon with 'n' sides is given by the formula:

Exterior Angle = 360° / n

Example:

A regular hexagon (n=6) has each exterior angle measuring 360°/6 = 60°.

4. Irregular Polygons

Irregular polygons have sides and angles of different measures. Finding exterior angles requires a more step-by-step approach:

Method: Using Supplementary Angles

• Step 1: Measure each interior angle.
• Step 2: For each interior angle, subtract its measure from 180° to find its corresponding exterior angle.

Example:

If an irregular pentagon has interior angles measuring 90°, 100°, 110°, 120°, and 140°, the corresponding exterior angles would be 90°, 80°, 70°, 60°, and 40°. Note that these exterior angles add up to 360°.

Applications of Exterior Angles

Understanding exterior angles has practical applications in various fields:

• Architecture and Construction: Exterior angles are crucial in designing structures with precise angles and ensuring stability.
• Cartography: Understanding angles is essential for creating accurate maps and surveying land.
• Engineering: Exterior angles are vital in calculating forces and stresses on structures.
• Computer Graphics: Exterior angles are used in creating and manipulating 3D models and shapes.
• Game Development: Exterior angles are used in pathfinding algorithms and other aspects of game design.

Solving Problems Involving Exterior Angles

Let's work through a few examples to solidify your understanding:

Problem 1:

A triangle has interior angles of 45°, 60°, and 75°. Find the measure of the exterior angle at the vertex with the 60° interior angle.

Solution:

The exterior angle is supplementary to the 60° interior angle, so it measures 180° - 60° = 120°. Alternatively, the sum of the remote interior angles (45° + 75°) is also 120°.

Problem 2:

A regular octagon has how many sides? What is the measure of each of its exterior angles?

Solution:

An octagon has 8 sides (n=8). The measure of each exterior angle is 360°/8 = 45°.

Problem 3:

An irregular quadrilateral has exterior angles of 75°, 85°, and 100°. Find the measure of the fourth exterior angle.

Solution:

The sum of the exterior angles of any quadrilateral is 360°. Therefore, the fourth exterior angle measures 360° - (75° + 85° + 100°) = 100°.

Conclusion

Finding exterior angles is a fundamental skill in geometry with numerous practical applications. By understanding the key properties, methods, and applications of exterior angles, you can effectively solve geometric problems and gain a deeper understanding of polygon properties. Remember the key relationships – supplementary angles with interior angles and the consistent 360° sum for convex polygons. This knowledge provides a solid foundation for more advanced geometric concepts. Practice is key to mastering these concepts, so work through various problems to solidify your understanding.