Lines and Angles

Mathematics has been an integral part of our everyday lives. This particular subject is the study of concepts like space, structure, change, and quantity. There are numerous sections and subsections and one of them is Geometry. Geometry is used to study the different shapes, sizes, spaces, and relative positions of a figure. They are all constituted with the help of points, lines, and angles that help to define their structures and areas easily. We will be going just a little deeper from the basic lines and angles introduction and other elementary topics like the class 3 line and point angles chapter. After having a surface-level discussion on the lines and angles introduction, we will see each of the two individually: introduction lines, and introduction to angles.


Introduction Lines

Introduction lines are defined as a collection of infinitesimal points. Having two arrowheads, a line can be extended in either direction. A segment, on the other hand, is a finite space between two defined points. It cannot be extended in either direction. It is a subset of the line and is used to measure its length. It has been observed students often confuse introduction lines with segments. The basic difference between the introduction lines and segment is that the former can be extended while the length of the latter is fixed. This difference can be seen in class 3 lines and point angles chapter.


Types of Lines

Depending on its use, a line has four types. They are as follows:

  • Vertical Line: It is a line that exists in a vertical position. In terms of direction, it runs from the North to the South. In simpler terms, it is a line running from the top of the page to the bottom of the page.

  • Horizontal Line: It is a line that exists in a horizontal position. In terms of direction, it runs from the East to the West. In simpler terms, it is a line running from the left side of the page to the right side of the page.

  • Parallel Line: When two lines run in the same direction without intersecting or meeting each other they are called parallel lines. These lines can be extended infinitesimally, yet they will never meet at any point.

  • Perpendicular Line: Two straight lines are said to be perpendicular to each other when they intersect each other at a 90-degree angle. The measure of their angle is also known as a right angle. The point at which they intersect is called their point of intersection. Two perpendicular lines form four right angles at their point of intersection.


With the lines and angles introduction, it became easy to build shapes of variable lengths.


What is a Ray?

A Ray is a part of the line. It has a fixed starting point and can be extended in only one direction. It has no endpoint. Rays play an important role in the introduction to angles. One of the interesting things about ray is that the mathematical notation of a ray requires the latter or the end letter in the front followed by the next point. For clarity, see the below diagram, and the notation that follows. 


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Now, in the above diagram you see a ray AB, but while expressing it in mathematical notation we write \[\overline{BA}\] with an arrow sign at the top. 


Introduction to Angles

An Angle is the space that is formed by two rays that have the same vertex or endpoint. The two rays can be extended in the direction of the pointed arrow at their end. 


Types of Angles

Depending on its use, an angle can have a number of types. Some of them are as follows:

  • Acute Angle: It is an angle made by two rays that are 0° to 90° apart from each other. The angle value is always less than 90°.

  • Right Angle: It is an angle made by two rays that are perpendicular to each other. The measure of this angle is always 90°.

  • Obtuse Angle: It is an angle made by two rays that are 90° to 180° apart from each other. The measure of this angle is always greater than 90°.

These are the three primary angles. Some of the other types of angles include;

  • Straight Angle: This is an angle whose measure is equal to 180°. It looks like a straight line and has both its arms pointing in opposite directions.

  • Reflex Angle: This is a type of angle whose value is always the greater one between the two possible angles made by a pair of rays. Its range lies between 180° to 360°.

  • Complete Angle: It is the angle that completes one turn in either clockwise or anti-clockwise direction and returns back to its starting position. The measure of this angle is 360°.

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The introduction to angles has made it easier to determine how far apart, angularly, two lines or rays are. Also, introduction to angles provides important insight into various properties of angles which will be discussed later through solved examples and FAQs.


Solved Examples

Example - 1. Look at the figure given below from class 3 line and point angles syllabus. Identify all the angles in it.


Solution: There are two types of angles present in the given figure. They are Acute angle and the Right Angle. ∠ACD and DCB are acute angles with a measure less than 90°. ∠ACB is the right angle.

 

Value of ∠DCB is 30°. Then, the measure of angle ACD is 90 - 30 = 60°. ∠ACD and ∠DCB are also complementary angles as their sum adds up to 90°.


Example - 2 From the figure illustrated below find the value of ∠MOP, if ∠MOQ or ∠y=100°.


Now sum of angles ∠MOP and ∠MOQ is 180°, using this piece of information, we can easily find the value of 2x or ∠MOP.


As, ∠MOP + ∠MOQ = 180°

So, 2x + 100 = 180°


On solving the above equation, you will get 2x or ∠MOP as 80°. 


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You can find more such examples in class 3 line and point angles chapter, which will help you to brush up the concept. Also, concepts discussed in class 3 line and point angles will stay with you in the next clases, as it forms the foundation.

FAQ (Frequently Asked Questions)

Question 1. What are Complementary and Supplementary Angles?

Answer: Value two angles that sums to 90° are called complementary angles. For example, if the measure of one angle is 50° then its complement angle would be of 90° - 50° = 40°. Thus, if the measure of one angle is known then the value of the other can easily be found out by subtracting the known angle from 90°.


Supplementary angles are the pair of angles whose sum adds to 180°. So, if the measure of one angle is known then the value of the other can be found out by subtracting the known angle from 180°.

Question 2. What are Adjacent and Vertically Opposite Angles?

Answer: A pair of angles that have the same vertex and one arm common to each other are called adjacent angles. The adjacent angles are side by side to each other. Hence, they should not overlap when being measured. They can share the vertex in equal measures or can be complementary to each other.


Vertically opposite angles are the angles that have the same vertex but are placed exactly opposite to each other. They are formed by the same two lines that intersect each other. As they are a replica of one another, the measure of their angles is always equal.