[Maths Class Notes] on Linear Pair Of Angles Pdf for Exam

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 

1. 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.

2. Slice of pizza is an example of linear angles. When we see the intersections of pizza, we can notice the linear pair angles.

3. 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.

4. An electric pole is also a real-life example of a linear pair of angles whose sum is equal to 180 degrees.

5. 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.

6. 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.

7. 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.

8. 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°.

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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.

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.