Question

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

Answer 1

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.