Prove that the bisectors of a pair of vertically opposite angles are in the same straight line.

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Solution

Let AB and CD intersect at a point O

Also, let us draw the bisectors OP and OQ of and.

Therefore,

And

We know that,and are vertically opposite angles. Therefore, these must be equal, that is:

We know that:

From (i)

From (ii)

This means, , and form a linear pair.

Hence, POQ forms a straight line.

Thus, we can say that the bisectors of a pair of vertically opposite angles are in the same straight line.

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