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Line and Angles Formula: Definition, Types, Properties

Line and Angles Formula serve as fundamental building blocks, forming the cornerstone of this mathematical discipline. A line is essentially an infinitely extended.

Anchal Singh14 Sept, 2023

In the realm of geometry, Line and Angles Formula serve as fundamental building blocks, forming the cornerstone of this mathematical discipline. A line is essentially an infinitely extended, one-dimensional entity that can be envisioned as a series of closely spaced points. For instance, when you draw a horizontal mark on a piece of paper, you are essentially depicting a line, characterized by its length and the potential for infinite extension in both directions.

On the other hand, angles emerge as geometric constructs created by the convergence of two rays at a shared endpoint, known as the vertex. These angles are typically quantified in degrees, a unit of measurement that can be applied using a protractor. In essence, angles are fundamental in understanding the relationships and properties of geometric shapes and figures. It's important to recognize that lines and angles are pervasive in the realm of geometry, serving as the underpinning for the exploration of various types of lines and angles, each with its unique characteristics and significance. So, let's delve further into the diverse array of lines and angles in this lesson.

Definition of Line and Angles

The notion of lines was originally introduced by ancient mathematicians to represent straight entities devoid of width or depth. While lines are relatively straightforward to grasp, an angle is a geometric shape that takes shape at the juncture of two line segments, lines, or rays when two rays intersect within the same plane. Lines: A line is a one-dimensional entity that extends infinitely in both directions, devoid of any width. It comprises an infinite array of closely packed points. Euclid described a line as a length without breadth. In a Cartesian plane, it can be represented by the linear equation ax + by = c. Take note of the line illustrated in the figure below.

Also Check - Probability formula

Rays: Rays are essentially lines originating at a single point (the starting point) and extending indefinitely in one direction without termination. When two rays are conjoined end to end, they give rise to an angle. They are typically represented as →PQ. Line Segments: When a line possesses two distinct endpoints, it is referred to as a line segment. The length of a line segment is measurable and is symbolically expressed as AB. Angles: The intersection of two rays at a particular point results in the formation of an angle. Angles are typically quantified in degrees and are represented by the degree symbol (∘). This symbol is a measure of rotation. An angle can assume values ranging from 0∘ to 360∘ and is denoted by the symbol ∠. Consider the figure below, which illustrates ∠ABC.

Types of Line and Angles

Geometry encompasses various types of lines and angles. Let's begin by exploring the diverse classifications of lines. Varieties of Lines Lines can be categorized into distinct types based on their inherent characteristics. The following table outlines the various classifications of lines.

Also Check - Conic Section Formula

Lines are fundamental elements in art, design, mathematics, and various other fields. They can be categorized into several types based on their characteristics and uses. Here are some common types of lines: Straight Line: A straight line is the shortest distance between two points. It has a consistent direction and does not curve.

Download PDF Line and Angles Formula

Curved Line: Curved lines are lines that change direction smoothly, forming curves or arcs. They can be gentle curves or more pronounced, like circles or spirals. Horizontal Line: A horizontal line is parallel to the horizon, and it runs from left to right. It suggests stability, rest, and tranquility. Vertical Line: A vertical line is perpendicular to the horizon and runs from top to bottom. It implies strength, stability, and potential for growth. Diagonal Line: Diagonal lines slant at an angle between horizontal and vertical. They add energy and movement to a composition and can suggest tension or action.

Types of Angles

Angles can be classified into various categories based on their measurements. There are six primary types of angles: Acute Angle: An angle is considered acute when its measurement is less than 90 degrees (90°). Obtuse Angle: An angle falls into the obtuse category if its measurement is greater than 90 degrees (90°) but less than 180 degrees (180°). Right Angle: A right angle is defined by an angle measuring exactly 90 degrees (90°). Straight Angle: When an angle measures precisely 180 degrees (180°), it is called a straight angle. Reflex Angle: An angle is termed reflex when its measurement exceeds 180 degrees (180°) but is less than 360 degrees (360°). Complete Angle: A complete angle is characterized by its measurement of 360 degrees (360°).

Properties of Line and Angles

As we have observed, there exist various types of lines and angles, each possessing unique characteristics. Nonetheless, there are some fundamental properties shared by lines and angles that warrant discussion. Properties of Lines: Definition: Lines can be defined as an infinite collection of points situated closely to one another. Infinite Extent: Lines extend infinitely in both directions without end. One-Dimensional: Lines are one-dimensional entities, lacking depth or thickness. Properties of Angles: Geometrical Formation: An angle is a geometrical shape that takes form when two lines intersect each other within the same plane. Arm and Vertex: The lines that intersect to create an angle are referred to as its arms or sides, and the point where they intersect is known as the vertex of the angle.

Tips on Line and Angles

Here is a compilation of key pointers to keep in mind when studying lines and angles:
Ubiquity in Geometry: Angles and lines are ubiquitous elements found in all geometric shapes.
Line Characteristics: A line is a one-dimensional entity, devoid of width, and stretches infinitely in both directions.
Infinite Lines: These are lines that originate from a specific starting point and extend infinitely in one direction. They serve as the building blocks for angle formation.
Angle Formation: Angles materialize when two rays intersect at a common point.

Examples on Line and Angles Formula

Example 1: Given that the measure of ∠AOB is 57°, let's find the measure of its reflex angle.

Solution:

According to the properties of lines and angles, we know that the sum of an angle and its reflex angle is always 360°. Therefore, ∠AOB + Reflex ∠AOB = 360° Reflex ∠AOB = 360° - 57° Reflex ∠AOB = 303° Answer: The reflex ∠AOB measures 303°. Example 2: In the provided figure where AOB forms a straight line, ∠AOC = 4x, and ∠BOC = 2x, we need to determine the value of x.

Solution:

One of the properties of lines and angles states that the sum of adjacent linear angles formed by a line is always 180°. Therefore, 4x + 2x = 180° 6x = 180° x = 180° / 6 = 30° So, we have found that x equals 30°. Consequently, the angle ∠AOB is 4x = 4 × 30 = 120°, and the angle ∠BOC is 2x = 2 × 30 = 60°. Answer: x = 30°, ∠AOB = 120°, ∠BOC = 60°.

Line and Angles Formula FAQs

Define the terms Line and Angles.

Line: A line can be described as an unbroken succession of closely positioned points that extends infinitely in both directions. It possesses just one dimension, which is its length. Angle: An angle is a geometric figure that takes shape when two rays, lines, or line segments intersect one another within the same plane. In simpler terms, an angle is formed when two rays, lines, or line segments meet at a common point.

Explain different types of Lines and Angles.

In geometry, various types of lines and angles exist. There are six primary angle types: right angles, acute angles, obtuse angles, straight angles, reflex angles, and complete angles. Additionally, lines in geometry can be categorized into different types, including horizontal lines, vertical lines, parallel lines, perpendicular lines, and transversal lines.

What is meant by Perpendicular Lines?

Perpendicular lines are a pair of straight lines that intersect at a right angle, meaning the angle formed at their point of intersection is exactly 90 degrees (90°). In other words, when two lines are perpendicular to each other, they create four right angles at the point of intersection.

Define the term vertical and horizontal lines.

Horizontal lines are lines that run parallel to the x-axis and are at right angles (perpendicular) to the y-axis. Vertical lines are lines that run parallel to the y-axis and are at right angles (perpendicular) to the x-axis.

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Definition of Line and Angles

Also Check - Probability formula

Types of Line and Angles

Also Check - Conic Section Formula

Download PDF Line and Angles Formula

Types of Angles

Properties of Line and Angles

Tips on Line and Angles

Here is a compilation of key pointers to keep in mind when studying lines and angles:
Ubiquity in Geometry: Angles and lines are ubiquitous elements found in all geometric shapes.
Line Characteristics: A line is a one-dimensional entity, devoid of width, and stretches infinitely in both directions.
Infinite Lines: These are lines that originate from a specific starting point and extend infinitely in one direction. They serve as the building blocks for angle formation.
Angle Formation: Angles materialize when two rays intersect at a common point.

Examples on Line and Angles Formula

Example 1: Given that the measure of ∠AOB is 57°, let's find the measure of its reflex angle.

Solution: