Which Of The Following Are Exterior Angles

Which of the Following Are Exterior Angles? A Comprehensive Guide

Understanding exterior angles is crucial for mastering geometry. This comprehensive guide will delve deep into the definition, identification, properties, and applications of exterior angles, helping you confidently differentiate them from interior angles and other geometric concepts. We'll explore various scenarios and examples to solidify your understanding.

What are Exterior Angles?

An exterior angle of a polygon is formed by extending one of its sides. It's the angle formed outside the polygon between the extended side and the adjacent side. Crucially, it's supplementary to its adjacent interior angle. This means that the exterior angle and its adjacent interior angle add up to 180 degrees.

Key characteristics of exterior angles:

• Location: Exterior angles are located outside the polygon.
• Formation: They are created by extending one side of the polygon.
• Supplementary relationship: Each exterior angle forms a linear pair with its adjacent interior angle, meaning their sum is 180°.
• Sum of exterior angles: The sum of the exterior angles of any polygon (convex or concave), when taking one exterior angle at each vertex, always equals 360°. This is a fundamental theorem in geometry.

Identifying Exterior Angles: Examples and Non-Examples

Let's clarify this with some examples and illustrations:

Example 1: A Triangle

Imagine a triangle ABC. If you extend side AB beyond point B, the angle formed outside the triangle adjacent to angle B is an exterior angle. Similarly, extending side BC beyond C and AC beyond A will create two more exterior angles. Each exterior angle has an adjacent interior angle (angles B, C, and A respectively).

Example 2: A Quadrilateral

Consider a quadrilateral with vertices W, X, Y, and Z. Extending side WX beyond X creates an exterior angle. Similarly, extending sides XY, YZ, and ZW beyond Y, Z, and W respectively will form three more exterior angles. Each of these exterior angles forms a linear pair with its adjacent interior angle.

Non-Examples:

• Interior Angles: Angles within the polygon are interior angles, not exterior angles.
• Angles Formed by Intersecting Diagonals: Angles formed by intersecting diagonals within the polygon are not exterior angles.
• Remote Interior Angles: While related to exterior angles, remote interior angles are the two angles inside the polygon that are not adjacent to the exterior angle in question. They are important in understanding the Exterior Angle Theorem, but they are not exterior angles themselves.

Exterior Angle Theorem

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. This theorem is a powerful tool for solving problems involving triangles.

Example:

In triangle ABC, if you extend side AB beyond B to create exterior angle DBC, then:

m∠DBC = m∠A + m∠C

This theorem highlights the relationship between exterior angles and interior angles, emphasizing that the exterior angle contains the combined measure of the two opposite interior angles.

The Sum of Exterior Angles of Any Polygon

A remarkable property of exterior angles is that their sum is always 360 degrees for any convex polygon. This holds true regardless of the number of sides. This fact is incredibly useful in various geometric problems.

Let's consider a few polygons:

• Triangle: A triangle has three exterior angles (one at each vertex), and their sum is always 360°.
• Quadrilateral: A quadrilateral also has four exterior angles, and their sum will always be 360°.
• Pentagon: A pentagon (five sides) will have five exterior angles, and their sum will still be 360°.
• N-sided polygon: For any polygon with 'n' sides, the sum of its exterior angles (taking one exterior angle at each vertex) is always 360°.

This consistent sum of 360 degrees for exterior angles is irrespective of the polygon's shape or size, making it a fundamental concept in geometry. This is unlike the sum of interior angles, which varies depending on the number of sides.

Exterior Angles and Concave Polygons

While the previous discussion focused on convex polygons (polygons where all interior angles are less than 180°), the sum of exterior angles theorem also applies to concave polygons (polygons with at least one interior angle greater than 180°). However, when dealing with concave polygons, you must be careful in selecting the exterior angle at each vertex. You must consistently choose the exterior angle that forms a linear pair with the interior angle, even if it means selecting the reflex angle. The sum will still equal 360 degrees.

Applications of Exterior Angles

Understanding exterior angles has numerous applications in various areas, including:

• Solving geometric problems: Exterior angles are frequently used to solve problems involving triangles and other polygons. By applying the Exterior Angle Theorem and the sum of exterior angles theorem, you can deduce unknown angle measures.
• Navigation and surveying: Exterior angles are used in surveying and navigation to calculate bearings and distances.
• Computer graphics and animation: Exterior angles are used in computer graphics to create realistic representations of objects and scenes.
• Architecture and engineering: Exterior angles play a role in architectural design and engineering calculations for building structures and infrastructure.

Distinguishing Exterior Angles from Other Angles

It's vital to distinguish exterior angles from other angles:

• Interior Angles: These are angles inside the polygon. They are supplementary to their adjacent exterior angles.
• Central Angles: These angles have their vertex at the center of a circle, and their rays pass through two points on the circle.
• Inscribed Angles: These angles are formed by two chords that intersect on a circle. They are half the measure of the intercepted arc.
• Vertical Angles: Formed by intersecting lines, these angles are not specifically associated with polygons.

Practice Problems

To reinforce your understanding, try these problems:

1. A triangle has interior angles of 70°, 60°, and 50°. Calculate the measure of each exterior angle.
2. A quadrilateral has exterior angles of 80°, 100°, 120°, and x°. Find the value of x.
3. A hexagon has five exterior angles measuring 60°, 70°, 80°, 90°, and 100°. Determine the measure of the sixth exterior angle.
4. In a triangle, one exterior angle measures 110°. What is the sum of the two remote interior angles?
5. Draw a concave quadrilateral and find the sum of its exterior angles. Remember to select exterior angles that are supplementary to the interior angles.

By working through these examples and problems, you will solidify your understanding of exterior angles, their properties, and their applications in various geometrical scenarios. Remember, consistent practice is key to mastering geometry concepts. Keep exploring, and you will become confident in identifying and utilizing exterior angles.