How To Find Measure Of Exterior Angle

Mastering Exterior Angles: A Comprehensive Guide to Finding Their Measure

Understanding exterior angles is crucial for mastering geometry. This comprehensive guide will walk you through everything you need to know about finding the measure of exterior angles, from basic definitions and theorems to advanced problem-solving techniques. We'll cover various polygon types and delve into the practical applications of this fundamental geometric concept. Whether you're a high school student tackling geometry homework or an adult brushing up on your math skills, this guide will equip you with the knowledge and confidence to conquer exterior angles.

I. Introduction to Exterior Angles

An exterior angle of a polygon is formed by extending one of its sides. Imagine taking one side of a triangle, square, or any other polygon and extending it outwards. The angle formed outside the polygon, between the extended side and the adjacent side, is the exterior angle. Crucially, each interior angle of a polygon has a corresponding exterior angle. Understanding the relationship between interior and exterior angles is key to solving problems.

II. The Exterior Angle Theorem

The foundation for calculating exterior angles lies in the Exterior Angle Theorem. This theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote (non-adjacent) interior angles. Let's break it down:

• Exterior Angle: The angle formed outside the triangle by extending one side.
• Remote Interior Angles: The two interior angles that are not adjacent to the exterior angle.

For example, in a triangle ABC, if side BC is extended to form an exterior angle at C, then the measure of that exterior angle is equal to the sum of the measures of angles A and B.

Mathematically: Exterior Angle = Remote Interior Angle 1 + Remote Interior Angle 2

This theorem provides a direct method for finding the measure of an exterior angle if you know the measures of the two remote interior angles.

III. Finding the Measure of Exterior Angles: Step-by-Step Guide

Let's explore how to find the measure of exterior angles in various scenarios:

A. Triangles:

1. Identify the Exterior Angle: Determine which exterior angle you need to find the measure of.

2. Identify the Remote Interior Angles: Locate the two interior angles that are not adjacent to the chosen exterior angle.

3. Add the Remote Interior Angles: Sum the measures of these two angles.

4. The Result is the Exterior Angle Measure: This sum is equal to the measure of the exterior angle.

Example:

In triangle XYZ, angle X = 50°, angle Y = 60°. If side YZ is extended to form an exterior angle at Z, what is the measure of this exterior angle?

1. Exterior angle: Angle at Z (exterior).

2. Remote interior angles: Angles X and Y.

3. Sum of remote interior angles: 50° + 60° = 110°

4. Therefore, the measure of the exterior angle at Z is 110°.

B. Quadrilaterals and Other Polygons:

The Exterior Angle Theorem doesn't directly apply to polygons with more than three sides. However, we can still find the measure of exterior angles using the following principles:

1. Sum of Exterior Angles: The sum of the exterior angles of any polygon (regardless of the number of sides) is always 360°. This is a fundamental property of polygons.

2. Regular Polygons: If the polygon is regular (meaning all sides and angles are equal), then you can find the measure of a single exterior angle by dividing 360° by the number of sides.

3. Irregular Polygons: For irregular polygons, you'll need to find the measure of the interior angles first, then use the fact that an interior and exterior angle are supplementary (they add up to 180°).

Example (Regular Polygon):

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

1. Number of sides: 6

2. Sum of exterior angles: 360°

3. Measure of each exterior angle: 360° / 6 = 60°

Example (Irregular Polygon):

In an irregular quadrilateral ABCD, angle A = 100°, angle B = 110°, angle C = 70°. Find the measure of the exterior angle at D.

1. Find angle D: The sum of interior angles in a quadrilateral is 360°. Therefore, angle D = 360° - (100° + 110° + 70°) = 80°

2. Exterior angle at D: Since interior and exterior angles are supplementary, the exterior angle at D = 180° - 80° = 100°.

IV. Explanation of the Exterior Angle Theorem (Scientific Perspective)

The Exterior Angle Theorem is a direct consequence of the properties of parallel lines and angles formed by transversals. Consider a triangle with an exterior angle. By drawing a line parallel to one of the sides through the vertex opposite the exterior angle, you can create corresponding angles and alternate interior angles. Using the properties of these angles (that corresponding angles are equal and alternate interior angles are equal), you can algebraically prove that the exterior angle is equal to the sum of the two remote interior angles. This rigorous proof demonstrates the theorem's mathematical foundation. It's a beautiful example of how seemingly unrelated geometric concepts connect to create elegant and powerful theorems.

V. Applications of Exterior Angles

Understanding exterior angles has several practical applications:

• Navigation: Exterior angles are used in surveying and navigation to determine directions and angles.

• Architecture and Construction: Exterior angles are essential in designing structures, ensuring proper angles and stability.

• Engineering: Many engineering designs rely on precise angular measurements, utilizing the properties of exterior angles.

• Computer Graphics: In computer graphics and game development, the calculation of exterior angles plays a crucial role in rendering and creating realistic 3D models.

VI. Frequently Asked Questions (FAQ)

Q: Can an exterior angle be greater than 180°?

A: No. Exterior angles are always less than 180°. They are formed by extending a side, creating an angle external to the polygon.

Q: What is the relationship between the interior and exterior angles of a polygon?

A: An interior angle and its corresponding exterior angle are supplementary; their sum is always 180°.

Q: Does the Exterior Angle Theorem apply to all polygons?

A: The Exterior Angle Theorem in its simplest form (sum of two remote interior angles) specifically applies to triangles. However, the principle that the sum of exterior angles is 360° applies to all polygons.

Q: How can I visualize exterior angles better?

A: Try drawing polygons and extending their sides. Use a protractor to measure the angles, both interior and exterior, and verify the relationships discussed in this article. Using interactive geometry software can also greatly improve your understanding.

Q: What if I don't know the measure of all interior angles in an irregular polygon?

A: If you only know some interior angles, you can use the formula for the sum of interior angles of a polygon ((n-2) * 180°, where n is the number of sides) to find the missing angle(s) before determining the exterior angles.

VII. Conclusion

Mastering the concept of exterior angles requires understanding the Exterior Angle Theorem for triangles and the universal property that the sum of exterior angles of any polygon is 360°. This knowledge is valuable not just for solving geometrical problems but also for comprehending the underlying principles of geometry and its various applications in different fields. By following the step-by-step guide and understanding the mathematical basis, you'll develop the skills to confidently tackle problems involving exterior angles, strengthening your foundation in geometry. Remember practice makes perfect; continue working through problems, and soon you’ll be a geometry expert!