Question

prove that if two lines intersect each other then vertically opposite angles are equal

Answer 1

Proof:

Note: Refer the image attached below

As per the problem we need to prove, $\angle \mathrm{AOD}=\angle \mathrm{BOC} \text { and } \angle \mathrm{AOC}=\angle \mathrm{BOD}$

Now on line AB,

$\angle \mathrm{AOD}+\angle \mathrm{BOD}=180^{\circ}$

On line CD,

$\angle \mathrm{BOC}+\angle \mathrm{BOD}=180^{\circ}$

Hence, $\angle \mathrm{AOD}+\angle \mathrm{BOD}=\angle \mathrm{BOC}+\angle \mathrm{BOD}$

$\angle \mathrm{AOD}=\angle \mathrm{BOC}$ --------------(Proved)

On line AB,

$\angle \mathrm{AOC}+\angle \mathrm{BOC}=180^{\circ}$

On Line CD,

$\angle \mathrm{BOC}+\angle \mathrm{BOD}=180^{\circ}$

Hence, $\angle \mathrm{AOC}+\angle \mathrm{BOC}=\angle \mathrm{BOC}+\angle \mathrm{BOD}$

$\angle \mathrm{AOC}=\angle \mathrm{BOD}$…………(Proved)

Hence the statement 'if two lines intersect each other then vertically opposite angles are equal' is proved.

Answer 2

Answer:

HEY MATE THIS IS YOUR ANSWER

Step-by-step explanation:

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