How To Find Exterior Angle

How to Find Exterior Angles: A Comprehensive Guide

Finding exterior angles might seem daunting at first, but with a clear understanding of geometric principles, it becomes a straightforward process. This comprehensive guide breaks down the concept of exterior angles, providing various methods to calculate them, tackling common challenges, and addressing frequently asked questions. Whether you're a student struggling with geometry or a curious learner seeking a deeper understanding, this guide will equip you with the knowledge and skills to master exterior angles. This article will cover polygons, focusing on triangles and quadrilaterals, and will explain the relationship between interior and exterior angles.

Understanding Exterior Angles

An exterior angle is the angle formed by one side of a polygon and the extension of an adjacent side. Crucially, it's outside the polygon. Each vertex of a polygon has two exterior angles, one on each side. We typically focus on one specific exterior angle at a given vertex. These angles are supplementary to their corresponding interior angles.

Interior and Exterior Angle Relationship

The relationship between interior and exterior angles is fundamental to solving problems involving exterior angles. They are supplementary, meaning that their sum is always 180°. This means that if you know the measure of an interior angle, you can easily find the measure of its corresponding exterior angle, and vice versa.

• Interior Angle + Exterior Angle = 180°

Methods for Finding Exterior Angles

The method for finding exterior angles varies depending on the type of polygon. Let's explore different approaches:

1. Using the Interior Angle:

This is the simplest method. If you know the measure of the interior angle, subtract it from 180°.

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

2. Using the Sum of Exterior Angles:

The sum of the exterior angles of any polygon (convex or concave) is always 360°. This is a powerful rule that allows us to find individual exterior angles if we know the number of sides.

• Formula: Individual Exterior Angle = 360° / Number of Sides

Example: A regular pentagon (5 sides) has exterior angles of 360° / 5 = 72°. Each exterior angle measures 72°. Note that this method only works reliably for regular polygons (polygons with all sides and angles equal).

3. Triangles:

Triangles offer a special case. The exterior angle of a triangle is equal to the sum of the two opposite interior angles. This is known as the Exterior Angle Theorem.

• Exterior Angle = Opposite Interior Angle 1 + Opposite Interior Angle 2

Example: Consider a triangle with interior angles measuring 60°, 70°, and 50°. The exterior angle at the vertex with the 50° interior angle will be 60° + 70° = 130°.

4. Quadrilaterals:

For quadrilaterals, while the sum of exterior angles remains 360°, there isn't a direct shortcut like the Exterior Angle Theorem for triangles. However, understanding the relationship between interior and exterior angles remains key. You'll need to utilize the sum of interior angles (360° for quadrilaterals) and the supplementary relationship between interior and exterior angles to solve for unknown exterior angles.

Step-by-Step Examples

Let's work through some detailed examples to solidify our understanding.

Example 1: Finding Exterior Angles of a Regular Hexagon

A regular hexagon has six equal sides and six equal angles.

1. Find the sum of exterior angles: The sum of exterior angles of any polygon is 360°.
2. Divide by the number of sides: 360° / 6 sides = 60°.
3. Result: Each exterior angle of a regular hexagon measures 60°.

Example 2: Finding an Exterior Angle Using the Interior Angle

An interior angle of a polygon measures 115°.

1. Use the supplementary relationship: 180° - 115° = 65°.
2. Result: The corresponding exterior angle is 65°.

Example 3: Applying the Exterior Angle Theorem (Triangles)

A triangle has interior angles of 45°, 65°, and 70°. Find the exterior angle at the vertex with the 45° interior angle.

1. Identify opposite interior angles: The opposite interior angles are 65° and 70°.
2. Add the opposite interior angles: 65° + 70° = 135°.
3. Result: The exterior angle is 135°.

Example 4: Solving for an Unknown Exterior Angle in a Quadrilateral

A quadrilateral has three exterior angles measuring 70°, 80°, and 100°. Find the fourth exterior angle.

1. Use the sum of exterior angles: The sum of exterior angles in any quadrilateral is 360°.
2. Sum known exterior angles: 70° + 80° + 100° = 250°.
3. Subtract from 360°: 360° - 250° = 110°.
4. Result: The fourth exterior angle measures 110°.

Solving Complex Problems

Some problems might involve more than one step. For instance, you might be given information about some interior angles and need to calculate the exterior angles before moving on to solve for other angles or sides. Remember to systematically use the relationships between interior and exterior angles, and the sum of exterior angles (360°), to solve these multi-step problems. Break down complex problems into smaller, manageable steps.

Common Mistakes to Avoid

• Confusing interior and exterior angles: Always clearly identify which angle you are working with.
• Incorrectly applying the Exterior Angle Theorem: This theorem only applies to triangles.
• Forgetting the sum of exterior angles is always 360°: This is a crucial rule for solving many problems.
• Miscalculating supplementary angles: Remember that interior and exterior angles are supplementary (add up to 180°).

Frequently Asked Questions (FAQ)

Q1: Can an exterior angle be greater than 180°?

No. An exterior angle is formed by extending one side of a polygon, so it cannot be greater than 180°. If you get a result above 180°, there's likely a calculation error.

Q2: What is the difference between an exterior angle and an exterior angle of a triangle?

The principle remains the same – both are angles formed outside the shape by extending a side. However, the Exterior Angle Theorem provides a unique shortcut for calculating exterior angles specifically within triangles.

Q3: Can a concave polygon have exterior angles?

Yes. The concept of exterior angles applies to both convex and concave polygons. However, for concave polygons, some exterior angles will be reflex angles (greater than 180° but less than 360°).

Q4: What if I have an irregular polygon?

For irregular polygons (polygons with unequal sides and angles), you can't use the shortcut of dividing 360° by the number of sides to find individual exterior angles. You need to work with the relationship between interior and exterior angles and the sum of interior angles to find unknown exterior angles.

Q5: How are exterior angles used in real-world applications?

Exterior angles are crucial in various fields including surveying, architecture, and engineering. They are essential for calculating angles and distances, ensuring the accuracy of construction plans, and navigating terrains.

Conclusion

Understanding how to find exterior angles is a fundamental skill in geometry. By mastering the concepts explained in this guide, you can confidently approach various problems involving polygons. Remember to leverage the relationship between interior and exterior angles, the sum of exterior angles (360°), and the Exterior Angle Theorem (for triangles) to solve problems efficiently. Practice is key – the more you work through examples, the more comfortable you’ll become with these concepts. Remember to break down complex problems into smaller, manageable steps. With consistent effort and a clear understanding of the underlying principles, you'll master the world of exterior angles in no time!