If two straight lines intersect each other, prove that the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.

#### Solution

Let AB and CD intersect at a point O

Also, let us draw the bisector *OP* of ∠AOC.

Therefore,

∠AOP = ∠POC (1)

Also, let’s extend *OP* to *Q*.

We need to show that, *OQ* bisects ∠BOD.

Let us assume that *OQ* bisects∠BOD, now we shall prove that *POQ* is a line.

We know that,

∠AOCand ∠DOBare vertically opposite angles. Therefore, these must be equal, that is:

∠AOC = ∠DOB (2)

∠AOPand ∠BOQ are vertically opposite angles. Therefore,

∠AOP = ∠BOQ

Similarly,

∠POC = ∠DOQ

We know that:

∠AOP +∠AOD+∠DOQ+∠POC+∠BOC+∠BOQ = 360°

2∠AOP+∠AOD+2∠DOQ+∠BOC =360°

2∠AOP + 2∠AOD+ 2∠DOQ = 360°

2(∠AOP+∠AOD+ ∠DOQ) = 360°

∠AOP+∠AOD+ ∠DOQ = `(360°)/2`

∠AOP+∠AOD+ ∠DOQ = 180°

Thus, POQ is a straight line.

Hence our assumption is correct. That is,

We can say that if the two straight lines intersect each other, then the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angles.