If two lines intersect, prove that the vertically opposite angles are equal.

Let us draw the figure.

Here, we can see that

AB and CD intersect each other at point E.

The two pairs of vertically opposite angles are -

1^st pair - ∠AEC and ∠BED

2^nd pair - ∠AED and ∠BEC

We need to prove that the vertically opposite angles are equal, i.e.,

∠AEC = ∠BED, and ∠AED = ∠BEC

Now, we can see that the ray AE stands on the line CD. We know, if a ray stands on a line then the sum of the adjacent angles is equal to 180°.

⇒ ∠AEC + ∠AED = 180° (By linear pair axiom) - - - - (i)

Similarly, the ray DE stands on the line AEB.

⇒ ∠AED + ∠BED = 180° (By linear pair axiom) - - - - (ii)

From equations (i) and (ii), we have

∠AEC + ∠AED = ∠AED + ∠BED

⇒ ∠AEC = ∠BED - - - - (iii)

Similarly, the ray BE stands on the line CED.

⇒ ∠DEB + ∠CEB = 180° (By linear pair axiom) - - - - (iv)

Also, the ray CE stands on the line AEB.

⇒ ∠CEB + ∠AEC = 180° (By linear pair axiom) - - - - (v)

From equations (iv) and (v), we have

∠DEB + ∠CEB = ∠CEB + ∠AEC

⇒ ∠DEB = ∠AEC - - - - (vi)

Thus, from equation (iii) and equation (vi), we have

∠AEC = ∠BED, and ∠DEB = ∠AEC

Therefore, it is proved that the vertically opposite angles are equal.