Question

Prove That: If two lines intersect each other, then the vertically opposite angles are congruent

Answer 1

Answer:

Theorem 6.1 : If two lines intersect each other, then the vertically opposite angles are equal.

Proof :

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) ∠ 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°? (Linear pair axiom)……………(2)

From (1) and (2), we can write∠ AOC + ∠ AOD = ∠ AOD + ∠ BOD

This implies that ∠ AOC = ∠ BOD

Similarly, it can be proved that ∠AOD = ∠BOC