Axioms

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. They lead to two pairs of vertically opposite angles, namely,

 (i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC.
We need to prove that ∠ AOC = ∠ BOD  and ∠ AOD = ∠ BOC.
Now, ray OA stands on line CD.
Therefore, ∠ AOC + ∠ AOD = 180°   (Linear pair axiom)                        (1)
Can we write ∠ AOD + ∠ BOD = 180°? Yes! (Why?)                               (2)
From (1) and (2), we can write
∠ AOC + ∠ AOD = ∠ AOD + ∠ BOD
This implies that ∠ AOC = ∠ BOD     (Refer Section 5.2, Axiom 3)
Similarly, it can be proved that ∠AOD = ∠BOC   
Now, let us do some examples based on Linear Pair Axiom and Theorem 6.1.