Find The Measure Of The Exterior Angle Shown

Find the Measure of the Exterior Angle Shown: A Comprehensive Guide

Understanding exterior angles is crucial in geometry, especially when dealing with polygons. This comprehensive guide will walk you through various methods to find the measure of an exterior angle, providing clear explanations, examples, and practice problems. We'll cover different polygon types and explore the relationships between interior and exterior angles. By the end, you'll confidently tackle any problem involving exterior angles.

Understanding Interior and Exterior Angles

Before diving into calculations, let's establish a solid understanding of the terminology.

Interior Angle: An interior angle is an angle inside a polygon, formed by two adjacent sides. Think of the angles within a triangle, square, or any other shape.

Exterior Angle: An exterior angle is formed by extending one side of a polygon. It's the angle between the extended side and the adjacent side. Crucially, an exterior angle and its adjacent interior angle are supplementary (they add up to 180°).

Key Relationship: The sum of the measures of the exterior angles of any polygon, one at each vertex, is always 360°. This is a fundamental property used extensively in solving problems.

Types of Polygons

Understanding the type of polygon is vital for calculating exterior angles. Some common types include:

• Triangle (3 sides): The simplest polygon.
• Quadrilateral (4 sides): Squares, rectangles, parallelograms, etc., all fall under this category.
• Pentagon (5 sides): A five-sided polygon.
• Hexagon (6 sides): A six-sided polygon.
• N-gon (n sides): A general term for a polygon with 'n' sides.

Methods for Finding Exterior Angles

Several methods can be used to determine the measure of an exterior angle, depending on the information given.

Method 1: Using the Supplementary Angle Property

This method is particularly useful when you know the measure of the adjacent interior angle.

Steps:

1. Identify the adjacent interior angle: Locate the interior angle that shares a side with the exterior angle you want to find.
2. Apply the supplementary angle rule: Remember that the sum of an interior angle and its adjacent exterior angle is 180°.
3. Calculate the exterior angle: Subtract the measure of the interior angle from 180°.

Example:

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

Method 2: Using the Sum of Exterior Angles

This method is powerful when dealing with the entire polygon, especially when the individual interior angles are unknown.

Steps:

1. Determine the number of sides (n): Count the number of sides of the polygon.
2. Apply the 360° rule: The sum of the exterior angles (one at each vertex) is always 360°.
3. Calculate the average exterior angle (for regular polygons): If the polygon is regular (all sides and angles are equal), divide 360° by the number of sides (n). This gives you the measure of each exterior angle.
4. For irregular polygons: This method alone won't give you the measure of a specific exterior angle, but it helps establish constraints and can be combined with other information.

Example (Regular Pentagon):

A regular pentagon has 5 sides. Therefore, each exterior angle measures 360°/5 = 72°.

Method 3: Using the Interior Angle Sum Formula (for regular polygons)

This approach is helpful when you know the number of sides and the polygon is regular.

Steps:

1. Find the sum of interior angles: The sum of interior angles of an n-sided polygon is given by the formula (n-2) * 180°.
2. Calculate the measure of each interior angle: For a regular polygon, divide the sum of interior angles by the number of sides (n).
3. Find the exterior angle: Subtract the interior angle from 180°.

Example (Regular Hexagon):

A regular hexagon has 6 sides. The sum of its interior angles is (6-2) * 180° = 720°. Each interior angle measures 720°/6 = 120°. Therefore, each exterior angle measures 180° - 120° = 60°.

Method 4: Using Trigonometry (for specific triangles)

In triangles, especially when dealing with lengths of sides, trigonometry can be employed to find exterior angles.

Steps:

1. Identify the relevant triangle: Focus on the triangle that includes the exterior angle you're interested in.
2. Apply sine, cosine, or tangent rules: Based on the information provided (side lengths and known angles), use appropriate trigonometric functions (sine rule, cosine rule) to find unknown angles within the triangle.
3. Determine the exterior angle: Use the supplementary angle property to find the exterior angle once you've found the adjacent interior angle using trigonometry.

Example:

Imagine a triangle with sides a, b, and c, and angles A, B, and C. If you know sides a and b, and angle C, you can use the cosine rule to find side c. Then, you can use the sine rule to find angles A and B. The exterior angle to angle A would be 180° - A.

Practice Problems

Let's solidify our understanding with some practice problems.

Problem 1:

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

Solution:

A regular octagon has 8 sides. The measure of each exterior angle is 360°/8 = 45°.

Problem 2:

An interior angle of a regular polygon measures 150°. How many sides does the polygon have?

Solution:

The exterior angle is 180° - 150° = 30°. Since the sum of exterior angles is 360°, the number of sides is 360°/30° = 12 sides.

Problem 3:

In a triangle ABC, angle A = 60°, and angle B = 75°. Find the measure of the exterior angle at vertex C.

Solution:

The sum of angles in a triangle is 180°. Therefore, angle C = 180° - 60° - 75° = 45°. The exterior angle at vertex C is 180° - 45° = 135°.

Problem 4:

A quadrilateral has interior angles measuring 70°, 100°, and 110°. Find the measure of the fourth interior angle and the exterior angle adjacent to it.

Solution:

The sum of interior angles in a quadrilateral is 360°. The fourth interior angle is 360° - 70° - 100° - 110° = 80°. The exterior angle adjacent to it is 180° - 80° = 100°.

Conclusion

Finding the measure of an exterior angle is a fundamental skill in geometry. By understanding the relationships between interior and exterior angles, and employing the methods outlined above, you can confidently solve various problems involving polygons. Remember to identify the type of polygon, utilize the supplementary angle property, and remember the crucial 360° rule for the sum of exterior angles. With practice, you'll become proficient in tackling these geometry challenges. Remember to always clearly define your approach and show your working to ensure accuracy and understanding.