Class 9 Maths Lines and Angles Pair of Angles

Pair of Angles

Some of the pair of angles we saw is below:

  1. Complimentary angles
  2. Supplementary angles
  3. Adjacent Angles
  4. Vertically opposite angles
  5. Linear Pair of Angles

 Now, let us find out the relation between the angles formed when a ray stands on a line. In the figure below ray QS stands on line PR. Notice that in this case ∠ PQR is a straight line  & thus sum of angles PQS & SQR  will be 180o.

 

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

Reverse Statement for this axiom: If the sum of two adjacent angles is 180°, then a ray stands on a line. Since the reverse statement is also true, we can have one more Axiom.

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

The two axioms above together is called the Linear Pair Axiom. It is called axiom, since there is no proof for this. It is accepted as a true statement based on observations.

 

Let us now examine the case when two lines intersect each other. We know that when two lines intersect, the vertically opposite angles are equal. Let’s try to prove it.

Theorem: If two lines intersect each other, then the vertically opposite angles are equal.

Since this is a theorem, we have to prove it. In case of Axiom, proof is not required; it is accepted as true statement based on observations.

Proof: Given Angle 1 & 3 are vertically opposite angle

To Prove: ∠ 1 = ∠ 3

Solution:

 ∠ 1 + ∠ 2 = 180o   (Using linear pair Axioms) --- (i)

∠ 3 + ∠ 2 = 180o   (Using linear pair Axioms) --- (ii)

Comparing equation (i) & (ii), we get

∠ 1 + ∠ 2 = ∠ 3 + ∠ 2

Or ∠ 1 = ∠ 3

 

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