Answer :

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.


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