Question

.......... Answer.........

Answer 1

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Answer 2

Solution-

In quadrilateral ABCD, sum of measurement of all interior angle must be equal to 360°

$\Rightarrow \angle A +\angle B+\angle C+\angle D=360$

And in triangle AOD, sum of measurement of all the angles must be equal to 180°

$\Rightarrow \angle OAD +\angle ODA+\angle AOD=180$

$\Rightarrow \angle OAD +\angle ODA=180-\angle AOD$

And,

$\angle A=2\times \angle OAD\ and\ \angle D=2\times \angle ODA$

As AO and DO are the angle bisectors.

Substituting the values,

$\Rightarrow \angle A +\angle B+\angle C+\angle D=360$

$\Rightarrow (2\times \angle OAD)+\angle B+\angle C+(2\times \angle ODA)=360$

$\Rightarrow \angle B+\angle C+2(\angle OAD+\angle ODA)=360$

$\Rightarrow \angle B+\angle C+2(180-\angle AOD)=360$

$\Rightarrow \angle B+\angle C+360-(2\times \angle AOD)=360$

$\Rightarrow \angle B+\angle C-2\angle AOD=0$

$\Rightarrow \angle B+\angle C=2\angle AOD\ \ (proved)$