6.4. Pairs of Angles

In Section 6.2, you have learnt the definitions of some of the pairs of angles such as complementary angles, supplementary angles, adjacent angles, linear pair of angles, etc. Can you think of some relations between these angles?

In this chunk, we focus on defining pairs of angles by referring back to the definitions learned previously. Pairs of angles include complementary angles (sum to 90°) and supplementary angles (sum to 180°). This forms the foundation for understanding their relationships, especially when a ray intersects a line.

Think of complementary angles like a pair of shoes that work perfectly together – together they create a complete look, just as these angles sum to 90°.

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 ∠AOC, ∠BOC and ∠AOB.

When a ray OC stands on a line AB at point O, it forms three angles: ∠AOC, ∠BOC, and ∠AOB. The important relationship here is that the sum of the two angles adjacent to the ray (AOC and BOC) equals the angle that includes them (AOB)—i.e., ∠AOC + ∠BOC = ∠AOB.

Imagine standing at a point where one path goes straight and another branches off. The angles formed as you look towards both paths relate to how much you've turned, similar to how the angles relate in our discussion.

Can we write ∠AOC + ∠BOC = ∠AOB? (1) Yes! (Why? Refer to adjacent angles in Section 6.2). What is the measure of ∠AOB? It is 180°. (Why?) (2) From (1) and (2), can you say that ∠AOC + ∠BOC = 180°? Yes! (Why?)

From the above discussion, we derive an important axiom: If a ray stands on a line, then the sum of the two adjacent angles so formed is 180°. This is established because ∠AOB is a straight angle that equals 180°.

Imagine a door opening perfectly in the middle of the wall; the angle from one side of the door to the other must add up to a straight line or 180°, just like the angles we discussed.

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?

You can also express the Converse of Axiom 6.1: If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a straight line. This establishes a two-sided understanding of the relationship between adjacent angles.

Threading a needle can help explain this. If you know how much thread comes through the eye (180°), you can assess the angles left over to guide the needle correctly.

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.

When two lines intersect, they form two pairs of vertically opposite angles that are equal. For example, if lines AB and CD intersect at O, then ∠AOC = ∠BOD and ∠AOD = ∠BOC. The equality of vertically opposite angles is an important concept often used in proofs.

Picture a ship's sails intersecting at the mast; the angles formed directly opposite each other are always the same, allowing for balanced sailing.

Now let us do some examples based on the Linear Pair Axiom and Theorem 6.1.

Various examples illustrate both the Linear Pair Axiom and Theorem involving vertically opposite angles. Solving these helps reinforce the concepts learned. For instance, you can imagine angles formed by intersecting roads and their relations.

Think of a traffic intersection; the angles at which cars turn relate closely to how we calculate their path, reflecting the principles of angle pairs.

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