Q. 54.2( 5 Votes )

# If two straight l

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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