Are you struggling to grasp the distinction between internal and external angles? In this article, we explain the definitions, properties, and applications of both types of angles. By understanding the relationship between internal and external angles, you’ll gain a deeper comprehension of geometric concepts. Let’s unravel the mystery behind internal and external angles together!
Key Takeaways
- Internal angles are formed within a polygon, while external angles are formed outside of it.
- The sum of all internal angles in a polygon is (n-2) times 180 degrees, where n is the number of sides or vertices.
- The sum of all external angles in a polygon is always 360 degrees.
- External angles are supplementary to their adjacent internal angles.
Definition of Internal Angle
An internal angle, also referred to as an interior angle, is a fundamental geometric concept defined within a polygon. In a polygon, which is a closed figure with straight line segments, an internal angle is formed at each vertex (corner) by two adjacent sides of the polygon.
It is the angle measured on the interior side of the polygon at the point where two sides meet. The sum of all internal angles in any polygon can be calculated using the formula (n-2) × 180 degrees, where n represents the number of sides in the polygon.
Definition of External Angle
An external angle, also known as an exterior angle, is a geometric term used in the field of mathematics and geometry. In a polygon, which is a closed figure made up of straight line segments, an external angle is formed at the vertex (corner) of the polygon by extending one of its sides.
More specifically, it is the angle between a side of the polygon and the line that extends from that side to the adjacent vertex outside the polygon. The external angle provides information about the direction and rotation when moving from one side of the polygon to another. The sum of all external angles in any polygon is always 360 degrees, regardless of the number of sides the polygon has.
Measurement of Internal Angles
When measuring internal angles, it is important to consider the contextually relevant polygon or triangle. The internal angle refers to the angle formed by two adjacent sides within the shape. To measure this angle, you can use a protractor or apply mathematical formulas depending on the type of polygon.
To measure the internal angles of a polygon, you can use the formula:
Internal Angle = (n-2) * 180° / n
- “n” represents the number of sides or vertices in the polygon.
- “(n-2)” is the number of triangles formed by connecting the vertices of the polygon.
- “180°” is the measure of a straight angle.
Here’s how you can calculate the internal angle of a polygon using this formula:
- Count the number of sides or vertices in the polygon.
- Subtract 2 from the number obtained in step 1 to find “(n-2),” which represents the number of triangles formed inside the polygon.
- Multiply “(n-2)” by 180° to calculate the total interior angle sum of the polygon.
- To find the measure of a single internal angle, divide the result from step 3 by “n”.
This formula applies to any polygon, from triangles (3 sides/vertices) to hexagons (6 sides/vertices), and beyond.
Measurement of External Angles
To measure the exterior angles of a polygon, you can use the exterior angle theorem. Here’s how to do it:
- Draw the Polygon: Start by drawing the polygon, making sure all the sides and vertices are clearly marked.
- Select an Exterior Angle: Choose one exterior angle that you want to measure. An exterior angle is formed by extending one side of the polygon beyond the vertex.
- Identify an Adjacent Interior Angle: Find the interior angle that is adjacent to the exterior angle you selected. This means it shares a vertex with the exterior angle and has one side in common with it.
- Apply the Exterior Angle Theorem: According to the exterior angle theorem, the measure of an exterior angle of a polygon is equal to the sum of the measures of its two adjacent interior angles. In mathematical terms:
Exterior Angle = Interior Angle 1 + Interior Angle 2
Or, if “x” represents the measure of the exterior angle:
x = Interior Angle 1 + Interior Angle 2
- Measure the Interior Angles: Use a protractor to measure the two adjacent interior angles.
- Calculate the Exterior Angle: Add the measures of the two adjacent interior angles to find the measure of the exterior angle.
Properties of Internal Angles
- Angle Sum Property: In any polygon, the sum of its internal angles is given by the equation: Internal Angle Sum = (n – 2) * 180°, where “n” represents the number of sides or vertices.
- Regular Polygons: In a regular polygon, all internal angles are congruent, meaning they have the same measure.
- Radians: Internal angle measures can also be expressed in radians, where one full revolution (360°) is equivalent to 2π radians.
- Linear Pair: A pair of internal angles that form a straight line, known as a linear pair, adds up to 180°.
- Exterior Angle Inequality Theorem: The measure of an exterior angle of a polygon is greater than the measure of any of its non-adjacent interior angles.
Properties of External Angles
Properties of external angles in geometry include:
- Angle Sum Property Theorem: The sum of an external angle and its corresponding internal angle in any polygon is always 180 degrees. This is a consequence of the angle sum property.
- Exterior Angle Theorem Formula: The exterior angle equation states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent interior angles. This can be represented as follows: Measure of exterior angle = Measure of interior angle 1 + Measure of interior angle 2.
- Reflex Angle: An exterior angle that measures greater than 180 degrees is called a reflex angle.
Knowing the properties of both internal and external angles will help you to answer questions when you have an unknown angle as you will be able to use this supplementary knowledge to calculate adjacent angles, alternate angles and congruent angles in simple polygons or in irregular polygons.
What are alternate interior angles?
Relationship Between Internal and External Angles
The relationship between internal and external angles in a polygon is governed by several key principles:
- Angle Sum Property: In any polygon, the sum of its internal angles is constant and equal to (n – 2) * 180 degrees, where “n” represents the number of sides or vertices of the polygon. This means that as the number of sides increases or decreases, the total sum of internal angles remains the same. For example, in a triangle (3 sides), the sum is always 180 degrees, and in a quadrilateral (4 sides), it is always 360 degrees.
- 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 non-adjacent interior angles. This relationship holds true for all polygons, not just triangles.
- Congruence: In regular polygons, all internal angles are congruent (have the same measure), and all external angles are also congruent. This congruence is a special property of regular polygons.
- Supplementary Angles: Internal angles and their corresponding external angles are supplementary, which means they add up to 180 degrees. This relationship is consistent in any polygon.
- Linear Pair: An internal angle and its corresponding external angle form a linear pair, which means they are supplementary and add up to 180 degrees.
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Frequently Asked Questions
What is the difference between an interior angle and an exterior angle?
An interior angle is formed within the polygon’s boundaries while an exterior angle is formed by extending one side of the polygon beyond the vertex.
How can I apply knowledge of internal and external angles in real-life situations or problem-solving?
Understanding these angles is essential in fields like architecture, engineering, and map-making, where angles and shapes play a crucial role in design and measurement.
Where can I find resources or tutoring to help me grasp the concepts of internal and external angles?
You can seek assistance from educational websites, textbooks, or tutoring services like Tutorax, which offer both in-person and online tutoring to accommodate your learning needs.