exterior angle

What is an Exterior Angle in Math? Definition

In mathematics, an exterior angle refers to the angle formed between a side of a polygon and the extension of its adjacent side. It is an angle that lies outside the polygon. Understanding exterior angles is crucial in geometry as it helps in determining various properties and relationships within polygons.

History of Exterior Angle

The concept of exterior angles has been studied for centuries. The ancient Greek mathematician Euclid, known as the "Father of Geometry," extensively discussed the properties of angles in his book "Elements" around 300 BCE. The understanding of exterior angles has evolved over time, with contributions from various mathematicians and scholars.

Grade Level for Exterior Angle

The concept of exterior angles is typically introduced in middle school or early high school mathematics. It is commonly covered in geometry courses, where students learn about polygons, angles, and their properties.

Knowledge Points of Exterior Angle and Detailed Explanation

To understand exterior angles, it is essential to have a grasp of the following knowledge points:

1. Polygon: A closed figure with straight sides, formed by connecting line segments.
2. Interior Angle: The angle formed inside a polygon at each vertex.
3. Exterior Angle: The angle formed outside a polygon at each vertex.
4. Adjacent Sides: The sides of a polygon that share a common vertex.

To calculate the measure of an exterior angle, follow these steps:

1. Identify the polygon and the vertex of interest.
2. Extend the adjacent side of the vertex to form a line.
3. Measure the angle formed between the extended side and the original side.

Types of Exterior Angle

There are two types of exterior angles:

1. Convex Exterior Angle: An exterior angle that measures less than 180 degrees.
2. Reflex Exterior Angle: An exterior angle that measures greater than 180 degrees.

Properties of Exterior Angle

The properties of exterior angles include:

1. The measure of an exterior angle is equal to the sum of the measures of its two remote interior angles.
2. The sum of all exterior angles of a polygon is always 360 degrees.

How to Find or Calculate Exterior Angle

To calculate the measure of an exterior angle, you can use the following formula:

Exterior Angle = 360 degrees / Number of Sides of the Polygon

Formula or Equation for Exterior Angle

The formula for calculating the measure of an exterior angle is:

Exterior Angle = 360 / n

Where "n" represents the number of sides of the polygon.

Application of the Exterior Angle Formula

The exterior angle formula is applied when you need to find the measure of an exterior angle in a polygon. By knowing the number of sides, you can easily calculate the measure using the formula mentioned above.

Symbol or Abbreviation for Exterior Angle

There is no specific symbol or abbreviation exclusively used for exterior angles. However, the term "ext. ∠" is often used to represent an exterior angle in mathematical notation.

Methods for Exterior Angle

There are several methods to solve problems related to exterior angles, including:

1. Using the exterior angle formula.
2. Applying the properties of exterior angles, such as the sum of exterior angles in a polygon.

Solved Examples on Exterior Angle

Example 1: Find the measure of each exterior angle of a regular hexagon. Solution: A regular hexagon has six sides. Using the formula: Exterior Angle = 360 / n Exterior Angle = 360 / 6 = 60 degrees Therefore, each exterior angle of a regular hexagon measures 60 degrees.

Example 2: In a pentagon, one exterior angle measures 45 degrees. Find the measure of each interior angle. Solution: The sum of an exterior angle and its corresponding interior angle is always 180 degrees. Let x be the measure of each interior angle. 45 degrees + x = 180 degrees x = 180 degrees - 45 degrees x = 135 degrees Therefore, each interior angle of the pentagon measures 135 degrees.

Example 3: Determine the measure of the reflex exterior angle in a heptagon. Solution: A heptagon has seven sides. Using the formula: Exterior Angle = 360 / n Exterior Angle = 360 / 7 ≈ 51.43 degrees Since the reflex exterior angle is greater than 180 degrees, it measures approximately 180 + 51.43 = 231.43 degrees.

Practice Problems on Exterior Angle

1. Find the measure of each exterior angle of a regular octagon.
2. In a quadrilateral, one exterior angle measures 120 degrees. Find the measure of each interior angle.
3. Determine the measure of the reflex exterior angle in a decagon.

FAQ on Exterior Angle

Question: What is an exterior angle? Answer: An exterior angle is the angle formed between a side of a polygon and the extension of its adjacent side.

In conclusion, understanding exterior angles is essential in geometry. It helps in analyzing polygons, calculating angles, and determining various properties. By applying the formula and properties of exterior angles, you can solve problems related to polygons effectively.