How To Find The Measure Of Each Exterior Angle

How to Find the Measure of Each Exterior Angle: A Comprehensive Guide

Understanding exterior angles is crucial in geometry, offering a pathway to solving complex problems involving polygons. This comprehensive guide will delve into the methods of finding the measure of each exterior angle, catering to various levels of understanding. We will cover the fundamental concepts, explore different approaches to problem-solving, and provide practical examples to solidify your grasp of this essential geometric concept.

Understanding Exterior Angles

Before diving into the methods, let's establish a clear understanding of what exterior angles are. An exterior angle is formed when a 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 each side of the extended line, but we typically focus on one at a time, usually the one that lies adjacent to the interior angle we're considering.

Key Properties of Exterior Angles:

• Supplementary Angles: 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 polygon (regardless of the number of sides) is always 360°. This property is fundamental to many calculations.
• Regular Polygons: In a regular polygon (where all sides and angles are equal), each exterior angle is equal to 360° divided by the number of sides. This provides a direct and simple method for finding the measure.

Methods for Finding the Measure of Each Exterior Angle

Several methods exist for determining the measure of each exterior angle, depending on the information provided. Let's examine the most common approaches:

Method 1: Using the Number of Sides (for Regular Polygons Only)

This method is the simplest and applies exclusively to regular polygons. The formula is:

Exterior Angle = 360° / Number of Sides

Example:

Find the measure of each exterior angle of a regular hexagon.

A hexagon has 6 sides. Using the formula:

Exterior Angle = 360° / 6 = 60°

Therefore, each exterior angle of a regular hexagon measures 60°.

Method 2: Using the Interior Angle (for Regular and Irregular Polygons)

This method works for both regular and irregular polygons. Since an exterior angle and its adjacent interior angle are supplementary, we can calculate the exterior angle if we know the interior angle. The formula is:

Exterior Angle = 180° - Interior Angle

Example:

The measure of an interior angle of a pentagon is 108°. Find the measure of its corresponding exterior angle.

Exterior Angle = 180° - 108° = 72°

Therefore, the corresponding exterior angle measures 72°. Note that this only gives us one exterior angle; if the pentagon is irregular, other exterior angles will have different measures.

Method 3: Using the Sum of Exterior Angles (for Irregular Polygons)

This method is particularly useful when dealing with irregular polygons where not all angles are known. We use the fact that the sum of all exterior angles is always 360°. While this doesn't directly give us the measure of each exterior angle, it allows us to find missing angles when some are already known.

Example:

An irregular quadrilateral has three exterior angles measuring 70°, 80°, and 110°. Find the measure of the fourth exterior angle.

Let the fourth exterior angle be 'x'. We know the sum of all exterior angles is 360°. Therefore:

70° + 80° + 110° + x = 360°

260° + x = 360°

x = 360° - 260° = 100°

The fourth exterior angle measures 100°.

Method 4: Using the Sum of Interior Angles (for Irregular Polygons)

This method also works well for irregular polygons. It combines the concept of supplementary angles with the formula for the sum of interior angles:

Sum of Interior Angles = (n - 2) * 180° where 'n' is the number of sides.

Once you've calculated the sum of interior angles, you can find each individual interior angle if they are all provided. Then, you can use Method 2 to find each corresponding exterior angle.

Example:

An irregular pentagon has interior angles measuring 100°, 110°, 120°, and 130°. Find the measure of each exterior angle.

1. Find the fifth interior angle: The sum of interior angles of a pentagon is (5-2) * 180° = 540°. The sum of the four given interior angles is 100° + 110° + 120° + 130° = 460°. Therefore, the fifth interior angle is 540° - 460° = 80°.

2. Find each exterior angle: Now, use Method 2 (Exterior Angle = 180° - Interior Angle) to find each exterior angle:

   • 180° - 100° = 80°
   • 180° - 110° = 70°
   • 180° - 120° = 60°
   • 180° - 130° = 50°
   • 180° - 80° = 100°

Advanced Applications and Problem Solving

The concepts of exterior angles extend beyond basic polygon calculations. They find applications in more complex geometric problems and other areas of mathematics.

Triangles and Exterior Angles

In triangles, the exterior angle theorem states that the measure of an exterior angle is equal to the sum of the two opposite interior angles. This theorem proves extremely useful in solving various triangle-related problems.

Tessellations and Exterior Angles

Exterior angles play a vital role in understanding tessellations (tilings of a plane). The sum of the exterior angles at any point in a tessellation must always equal 360°. This constraint helps determine which polygons can tessellate.

Solving Complex Geometric Problems

Understanding exterior angles enables us to solve more intricate problems involving multiple polygons, intersecting lines, and angle relationships. The ability to combine this knowledge with other geometric theorems allows us to break down complex problems into manageable steps.

Practical Exercises

To reinforce your understanding, try these exercises:

1. Find the measure of each exterior angle of a regular octagon.
2. An irregular hexagon has five exterior angles measuring 50°, 60°, 70°, 80°, and 90°. Find the measure of the sixth exterior angle.
3. A triangle has interior angles measuring 45° and 60°. Find the measure of the exterior angle adjacent to the third interior angle.
4. Can a regular pentagon tessellate a plane? Justify your answer using the exterior angle concept.

By working through these exercises and applying the various methods described, you'll develop a strong grasp of how to find the measure of each exterior angle in a variety of polygons. Remember, practice is key to mastering this crucial geometric concept.

Conclusion

Finding the measure of each exterior angle is a fundamental skill in geometry. Whether dealing with regular or irregular polygons, understanding the relationships between exterior and interior angles, and the constant sum of 360° for exterior angles, allows you to solve a wide range of problems. By mastering these techniques, you build a strong foundation for more advanced geometric concepts and problem-solving. Remember to carefully consider the properties of the polygon in question and choose the most efficient method for calculating the exterior angles. Through consistent practice and application, you'll become proficient in this essential geometric skill.