Mathematics

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


SOLUTION
AB and CD are straight lines intersecting at O. OX the bisector of angles $$\angle$$AOC and OY is the OY is the bisector of $$\angle$$BOD.

OY is the bisector of $$\angle$$BOD.

$$\therefore \angle 1=\angle 6$$ ……..$$(1)$$

OX is the bisector of $$\angle$$AOC

$$\therefore \angle 3=\angle 4$$ …….$$(2)$$

$$\angle 2=\angle 5$$ ……….$$(3)$$ (vertically opposite angles)

We know that, the sum of the angles formed at a point is $$360^o$$.

$$\therefore \angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6=360^o$$

$$\Rightarrow \angle 1+\angle 2+\angle 3+\angle 3+\angle 2+\angle 1=360^o$$
(using $$1, 2$$ and $$3$$)

$$\Rightarrow 2\angle 1+2\angle 2+2\angle 3=360^o$$

$$2(\angle 1+\angle 2+\angle 3)=360^o$$

$$\Rightarrow \angle DOY+\angle AOD+\angle AOX=180^o$$

$$\angle XOY=180^o$$

$$\therefore$$ The bisectors of pair of vertically opposite angles are on the same straight line.
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