Know About Linear Pair of Angles
Two angles are said to be linear angles if they are adjacent angles and are formed by two intersecting lines. The measure of a straight angle is 180 degrees, so the pair of linear angles must add up and form up to 180 degrees. Their noncommon sides form a straight line.
When two lines intersect each other at a common point then, a linear pair of angles are formed. If the angles are adjacent to each other after the intersection of the lines, then the angles are said to be adjacent. The sum of the linear pair of angles is always equal to 180 degrees. Such angles are also known as supplementary angles. The adjacent angles are the angles that have a common vertex. Hence, the linear pair of angles always have a common vertex. Also, there is a common arm that represents both the angles of the linear pair. A real-life example of a linear pair is a ladder that is placed against a wall, forming linear angles at the ground.
Basically, a linear pair of angles always lie on a straight line. The pair of adjacent angles are constructed on a line segment, but not all adjacent angles are linear. Hence, we can also say that a linear pair of angles is the adjacent angle whose non-common arms are basically opposite rays.
A ladder placed against the wall is a real-life example of linear angles and you will find many problems related to this real-life example in your books as well. Some of the real-life examples of linear angles are given below.
Real-Life Examples of Linear Angles
The ladder placed against the wall is a real-life example of a linear pair. Both the angles that are formed on the adjacent sides have a common vertex and their sum is equal to 180 degrees.
Slice of pizza is an example of linear angles. When we see the intersections of pizza, we can notice the linear pair angles.
Hands of the clock are a real-life example of linear angles. The hands of the clock are adjacent to each other, they have a common vertex and their sum of angles is 180 degrees.
An electric pole is also a real-life example of a linear pair of angles whose sum is equal to 180 degrees.
Scissors that are used in our daily life are classic daily life examples of linear angles. The flanks of scissors that are adjacent to each other have a common vertex, and the sum is equal to 180 degrees.
T function in the highway, that is on the roads, is a live example of linear angles. Whenever you hit the drive, don't forget to notice it.
Have you seen the justice balance? The one that is used in the courtroom, is the most common example of linear angles. The angles formed in the justice balance sum is 180 degrees.
Chopping board, daily used by our mothers in the kitchen. The sum of their angle is equal to 180 degrees. It is a pair of linear angles.
Explanation for Linear Pair of Angle
A straight angle is formed when the angle between two lines is 180 degrees. A straight line can be represented by using a straight angle. A circle of the infinite radius can be visualised on the straight line. The line segment is any part of the line having two endpoints. Also, the ray is that part of the line which has only one endpoint. A line segment with A and B as two endpoints is represented as AB.
If there is a pair of adjacent angles, then the pair is called a linear angle only if their sum is equal to 180 degrees. For example, the linear pair of 40 degrees and 140 degrees will sum up to 180 degrees and thus, these adjacent angles will be known as linear angles.
The line segment AB and two arrows at the end indicates a line is represented in the figure given below.
If a point O is taken anywhere on the line segment AB as shown, then the angle between the two line segments AO and OB is a straight angle i.e., 180°.
(Image will be Uploaded Soon)
Properties of Linear Pair of Angles
There are some unique properties of linear pairs of angles that make them unique and different from other pairs of angles. There are some properties of linear pair of angles and they are listed below:
The sum of two angles in the linear pair is always 180 degrees. Two adjacent angles, if not equal to 180 degrees, are not linear angles.
Linear pairs of angles always form in a straight line
Remember, all linear pair I'd angles are adjacent angles but all adjacent angles are not linear pair angles.
They can be considered as two parts of a straight angle or 180- degree angle.
Linear pairs of angles share a common vertex and also the common arm between them is shared.
Linear Pair Postulate
The postulate for linear pairs states that if a ray stands on the line then it is definitely that their sum is going to be equal to 180 degrees. If the sum of a pair of adjacent angles is 180 degrees then the common arm of two angles will form a line. Remember that linear pairs of angles are not always congruent.
Consider a Ray OP Stand on the Line Segment:
The angles which are formed at E are ∠QEM and ∠QEN. It is known that the angle between the two line segments ME and EN is 180°. Therefore, the angles ∠QEN and ∠QEM add up to 180°.
Thus, ∠QEM + ∠QEN = ∠MEN = 180°
∠QEN and ∠QEM are adjacent to each other, and when the sum of adjacent angles is 180°, then such angles form a linear pair of angles.
The above discussion can be stated as an axiom.
Axioms
Axiom 1: The adjacent angles form a linear pair of angles if a ray stands on a line.
In the figure given above, all line segments are passing through the point O, as shown in the figure. As the ray OF lies on the line segment MN, angles ∠FON and ∠FOM form a linear pair. Similarly, ∠GON and ∠HON form a linear pair and so on.
The converse of the stated axiom is also true, which can also be stated as the following axiom.
Axiom 2: If a linear pair is formed by two angles, the uncommon arms of the angles form a straight line.
Image will be uploaded soon
In the figure given above, only the last pair represents the linear pair, as the sum of two adjacent angles is 180°. Therefore, AB represents a line. The other two pairs of angles are adjacent, but they are not forming a linear pair. They do not form a straight line.
The two axioms mentioned above form Linear Pair Axioms and are helpful in solving various mathematical problems.
Questions:
Question 1: Find the values of the angles l, p, and q in each of the following questions
Solution:
(i) Lines AD and EC intersect
∴ ∠DOC = ∠AOE(vertically opposite angles)
Z = 40°
Now,
AD is a line
∠DOE + ∠AOE = 180°(Linear Pair)
p + 40° = 180°
p= 180° - 40°
p = 140°
Also, lines AD and CE intersect
∠DOE = ∠COA (vertically opposite)
p = ∠COB + ∠BOA
140° = l + 25°
140° - 25° = l
115° = l
p = 115°
(ii) Here,∠BOC = ∠AOD (vertically opposite angles)
l = 55°
Now, BD is a line
∠AOD + ∠AOB = 180°(linear pair)
55° + p = 180°
p = 180° - 55°
p = 125°
And,
∠AOB = ∠COD(vertically opposite angles)
p = q
125° = q
q = 125°
Question 2: In the adjoining figure:
(i) Is ∠1 adjacent to ∠2?
(ii) Is ∠AOC adjacent to ∠AOE?
(iii) Does the angles ∠COE and ∠EOD form a linear pair?
(iv) Are ∠BOD and ∠DOA supplementary?
(v) Is ∠1 vertically opposite to ∠4?
(vi) Find the vertically opposite angle of ∠5?
Solution:
(i) Here,
∠1 and ∠2 Have a common vertex O, They have a common line OC and There is no overlapping of angles.
∴ They are adjacent angles.
(ii) Here,
Angles are overlapping with each other.
∴ They are not adjacent angles.
(iii) ∠COE and ∠EOC have Common vertex O,
Common side OE
Their uncommon side COD forms a line.
∴ They form a linear pair.
(iv) ∠BOD and ∠DOC have Common vertex O,
Common side OE
Their uncommon side COD forms a line.
∴ They form a linear pair.
Since they are a linear pair
Their sum is 180°
∴ ∠BOD + ∠BOC = 180°
Since the sum is 180°
∴ They are supplementary angles.
(v) Here,∠1 and ∠4 have
A common vertex O,are opposite to each other.
∴Yes, ∠1 is vertically opposite to ∠4
(vi) Here, Two interesting lines are AB and CD and angle vertically opposite to ∠5 is ∠BOC
∴ ∠BOC is vertically opposite to ∠5.
FAQs on Linear Pair Of Angles
1. Can three angles form a linear pair?
Linear pair of angles consist of those angles whose sum is going to be 180 degrees, that is the supplementary angles lie on the common arm. If two angles form a linear pair, then angles are supplementary. A linear pair forms a straight angle of 180 degrees, so you have two angles whose sum is given as 180, which means they are supplementary. Hence vertical angles are not adjacent. That ∠1 and ∠3 are not vertical angles(they are a linear pair).
2. Are all adjacent angles linear pairs?
Angles that are next to each other are known as adjacent angles, i.e., two angles with one common arm. All adjacent angles are not linear, but all linear pairs are adjacent. In a linear pair, the arms of the angles that are not always common or collinear i.e., they lie on a straight line. Linear angles are adjacent angles but for being a linear pair they need to be supplementary, that is, their sum should be equal to 180 degrees and along with this, they should lie on a common arm.
3. How do you find a linear pair of angles?
If there are two angles, that is the pair of angles whose sum measures 180 degrees. So these pairs of angles add up to form 180 degrees. For example, for 160 degrees and 20-degree, the sum is 180 degrees if they are adjacent lying on a common arm or vertex. They are the ones who have the property of lying on a common arm. This is the most important topic for you to revise for exams.
4. From where can I get the notes for the linear pair topic?
You can easily get the notes of the topic in linear pairs from the Vedantu App or Vedantu website. Just by simply clicking on the option, you are going to come across the article that will provide you brief information about linear angles and along with it, this is going to help you to get better information on the topic and score well in the examination. These will clear your concept and help you to understand the topic in brief.
5. Why should I take notes from the Vedantu app?
The Vedantu app provides you with the best material designed by the experts for your help. These are going to help you to increase your ability and performance in your exams. Vedantu has designed the notes, especially from the experts of the particular subjects, so that it will be easier for the students to gain the knowledge and score well in the examinations. At Vedantu, our main focus is to provide you with the learning resources that will help you to get clarity of the topics that you have studied.