Linear Pair Of Angles


Linear pair is a pair of adjacent angles whose non- common 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.

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 visualized 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.

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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Consider a Ray OP Stand on the Line Segment  as Shown:

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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.

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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 forms a straight line.

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Figure 3 Adjacent Angles with Different Measures

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

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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?

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  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.


FAQ (Frequently Asked Questions)

1. Can three Angles form a Linear Pair?

A: 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?

A: 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.