If two straight line intersect each other, then the measurement of vertically opposite angles are equal – prove it logically.

Let us consider two straight lines PQ and RS intersecting at a point O.

Let ∠POR = ∠1

∠ROQ = ∠2

∠SOQ = ∠3

∠SOP = ∠4

Let us first find the pairs of vertically opposite angles from above diagram.

∠1 and ∠3 are lying opposite to each other [in common language lying face to face to each other].

So, first pair of vertically opposite angles is ∠1 and ∠3.

∠4 and ∠2 are also lying opposite to each other.

Hence ∠4 and angle 2 forms our second pair of vertically opposite angles.

To Prove: Vertically opposite angles are equal

Proof:

Now when we observe the above diagram, ray RO is perpendicular to line PQ.

So ∠1 + ∠2 = 180°……… (i) [As they form linear pair that is sum of angles in a straight line is 180°]

Similarly ray QO is perpendicular to line SR.

So ∠2 + ∠3 = 180°……… (ii) [As they form linear pair that is sum of angles in a straight line is 180°]

Now for equation (i) and (ii), the right-hand side is same. So, both the terms of left-hand side have to be equal.

That is ∠1 + ∠2 = ∠2 + ∠3

∠2 is eliminated and the final equation becomes,

∠1 = ∠3.

Hence it is proved that ∠1 and ∠3 are equal and at the same time they are vertically opposite angles.

Now ray PO is perpendicular to line SR.

So ∠4 + ∠1 = 180°……… (iii) [As they form linear pair that is sum of angles in a straight line is 180°]

Now for equation (i) and (iii), the right-hand side is same. So, both the terms of left-hand side has to be equal.

That is ∠1 + ∠2 = ∠4 + ∠1

∠1 is eliminated and the final equation becomes,

∠2 = ∠4.

Hence it is proved that ∠2 and ∠4 are equal and at the same time they are vertically opposite angles.