As we've learnt the definitions of some of the pairs of angles such as complementary angles, supplementary angles, adjacent angles, linear pair of angles, etc.
Now, let us find out the relation between the angles formed when a ray stands on a line. Draw a figure in which a ray stands on a line as shown in Fig. 6.6.
Name the line as AB and the ray as OC. What are the angles formed at the point O? They are ` angle AOC, angle BOC` and `angle AOB`. Can we write `angle AOC + angle BOC = angle AOB` ? (1) Yes! (Why? Refer to adjacent angles in Section 6.2 ) What is the measure of `angle AOB` ? It is `180°`.
From (1) and (2), can you say that `angle AOC + angle BOC = 180°` ? Yes ! (Why ?) From the above discussion, we can state the following Axiom :
`color {blue} text(Axiom 6.1 : If a ray stands on a line, then the sum of two adjacent angles so formed is 180°)`.
Recall that when the sum of two adjacent angles is `180°`, then they are called a linear pair of angles.
In Axiom 6.1, it is given that ‘a ray stands on a line’. From this ‘given’, we have concluded that ‘the sum of two adjacent angles so formed is `180°’.` Can we write Axiom 6.1 the other way? That is, take the ‘conclusion’ of Axiom 6.1 as ‘given’ and the ‘given’ as the ‘conclusion’. So it becomes:
(A) If the sum of two adjacent angles is `180°`, then a ray stands on a line (that is, the non-common arms form a line).
Now you see that the Axiom 6.1 and statement (A) are in a sense the reverse of each others. We call each as converse of the other. We do not know whether the statement (A) is true or not.
Let us check. Draw adjacent angles of different measures as shown in Fig. 6.7. Keep the ruler along one of the non-common arms in each case. Does the other non-common arm also lie along the ruler?
You will find that only in Fig. 6.7 (iii), both the non-common arms lie along the ruler, that is, points A, O and B lie on the same line and ray OC stands on it. Also see that `angle AOC + angle COB = 125° + 55° = 180°`. From this, you may conclude that statement (A) is true. So, you can state in the form of an axiom as follows:
Axiom 6.2 : If the sum of two adjacent angles is `180°`, then the non-common arms of the angles form a line.
For obvious reasons, the two axioms above together is called the Linear Pair Axiom.
Let us now examine the case when two lines intersect each other.
Recall, from earlier classes, that when two lines intersect, the vertically opposite angles are equal. Let us prove this result now. See Appendix 1 for the ingredients of a proof, and keep those in mind while studying the proof given below.
Theorem 6.1 : If two lines intersect each other, then the vertically opposite angles are equal.
Proof : In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,
(i) `angle AOC` and `angle BOD ` (ii) `angle AOD` and
`angle BOC.`
We need to prove that `angle AOC = angle BOD`
and `angle AOD = angle BOC`.
Now, ray `OA` stands on line `CD`.
Therefore, `angle AOC + angle AOD = 180°` (Linear pair axiom) (1)
Can we write `angle AOD + angle BOD = 180°`? Yes! (Why?) (2)
From (1) and (2), we can write
`angle AOC + angle AOD = angle AOD + angle BOD`
This implies that `angle AOC = angle BOD` (Refer Section 5.2, Axiom 3)
Similarly, it can be proved that `angle AOD = angle BOC`
Now, let us do some examples based on Linear Pair Axiom and Theorem 6.1.
As we've learnt the definitions of some of the pairs of angles such as complementary angles, supplementary angles, adjacent angles, linear pair of angles, etc.
Now, let us find out the relation between the angles formed when a ray stands on a line. Draw a figure in which a ray stands on a line as shown in Fig. 6.6.
Name the line as AB and the ray as OC. What are the angles formed at the point O? They are ` angle AOC, angle BOC` and `angle AOB`. Can we write `angle AOC + angle BOC = angle AOB` ? (1) Yes! (Why? Refer to adjacent angles in Section 6.2 ) What is the measure of `angle AOB` ? It is `180°`.
From (1) and (2), can you say that `angle AOC + angle BOC = 180°` ? Yes ! (Why ?) From the above discussion, we can state the following Axiom :
`color {blue} text(Axiom 6.1 : If a ray stands on a line, then the sum of two adjacent angles so formed is 180°)`.
Recall that when the sum of two adjacent angles is `180°`, then they are called a linear pair of angles.
In Axiom 6.1, it is given that ‘a ray stands on a line’. From this ‘given’, we have concluded that ‘the sum of two adjacent angles so formed is `180°’.` Can we write Axiom 6.1 the other way? That is, take the ‘conclusion’ of Axiom 6.1 as ‘given’ and the ‘given’ as the ‘conclusion’. So it becomes:
(A) If the sum of two adjacent angles is `180°`, then a ray stands on a line (that is, the non-common arms form a line).
Now you see that the Axiom 6.1 and statement (A) are in a sense the reverse of each others. We call each as converse of the other. We do not know whether the statement (A) is true or not.
Let us check. Draw adjacent angles of different measures as shown in Fig. 6.7. Keep the ruler along one of the non-common arms in each case. Does the other non-common arm also lie along the ruler?
You will find that only in Fig. 6.7 (iii), both the non-common arms lie along the ruler, that is, points A, O and B lie on the same line and ray OC stands on it. Also see that `angle AOC + angle COB = 125° + 55° = 180°`. From this, you may conclude that statement (A) is true. So, you can state in the form of an axiom as follows:
Axiom 6.2 : If the sum of two adjacent angles is `180°`, then the non-common arms of the angles form a line.
For obvious reasons, the two axioms above together is called the Linear Pair Axiom.
Let us now examine the case when two lines intersect each other.
Recall, from earlier classes, that when two lines intersect, the vertically opposite angles are equal. Let us prove this result now. See Appendix 1 for the ingredients of a proof, and keep those in mind while studying the proof given below.
Theorem 6.1 : If two lines intersect each other, then the vertically opposite angles are equal.
Proof : In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,
(i) `angle AOC` and `angle BOD ` (ii) `angle AOD` and
`angle BOC.`
We need to prove that `angle AOC = angle BOD`
and `angle AOD = angle BOC`.
Now, ray `OA` stands on line `CD`.
Therefore, `angle AOC + angle AOD = 180°` (Linear pair axiom) (1)
Can we write `angle AOD + angle BOD = 180°`? Yes! (Why?) (2)
From (1) and (2), we can write
`angle AOC + angle AOD = angle AOD + angle BOD`
This implies that `angle AOC = angle BOD` (Refer Section 5.2, Axiom 3)
Similarly, it can be proved that `angle AOD = angle BOC`
Now, let us do some examples based on Linear Pair Axiom and Theorem 6.1.