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CAT Linear Pair of Angles - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

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  • 3 Questions around this concept.

Solve by difficulty

On the figure, if AOB is a straight line, then x is 

 

Concepts Covered - 1

Linear Pair of Angles

Linear Pair of Angle: Two adjacent angles form a linear pair of angles if their non-common arms are two opposite rays, i.e., the sum of two adjacent angles is 180°.

In the above figure, ∠AOC and ∠BOC form a linear pair of angles.

Axiom 1: If a ray stands on a line, then the sum of two adjacent angles so formed is 180°. 

Let's see, how this is possible.

We have given A ray CD stands on a line AB such that ∠ACD and ∠BCD are formed. (As said in axiom). Now draw CE perpendicular to AB.

                ∠ACD = ∠ACE + ∠ECD               ...(i)
and          ∠BCD = ∠BCE - ∠ECD                ...(ii)
Adding (i) and (ii), we get
                ∠ACD +  ∠BCD = (∠ACE + ∠ECD) + (∠BCE - ∠ECD)
                                          = ∠ACE + ∠BCE
                                          = (90° + 90°) = 180°                 [As  ∠ACE = ∠BCE = 90°] 
Therefore, ∠ACD +  ∠BCD = 180°
 

Axiom 2 (Converse of Axiom 1): If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.

In the axiom above, it is giiven that, two adjacent angles AOC and BOC with common arm OC such that AOC + BOC = 180°. So we need to see if OA and OB are in the same straight line, i.e., AOB is a straight line.

Let's assume that, AOB is not a straight line. Then, produce AO to D so that AOD is a straight line. 

By our assumption, AOD is a straight line and ray OC lies on it.

Therefore,      AOC + COD = 180°         [Linear pair]
But,                AOC + BOC = 180°         [Given]
Therefore,      AOC + COD = AOC + BOC         [Each equal to 180°]
So,                               COD = BOC
But, this is not true, since a part cannot be equal to whole.
So, our assumption is wrong.
Hence, AOB is a straight line.

For obvious reasons, the two axioms above together is called the Linear Pair Axiom.

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