Question

Write the antecedent and consequentnes 'If two li intersect, then the vertically opposite angles are congruent'. ​

Answer 1

Step-by-step explanation

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

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

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