If two lines intersect prove that the vertically opposite angles are equal

Explanation

Given Two lines AB and CD intersect at point O.
To prove
(i) `" "/_AOC= /_`AOD
(ii)` " "/_AOD= /_`BOC Proof
(i) Since, ray OA stands on line CD
` :. " "/_AOC= /_AOD =180^(@) " " `[linear pair axiom].....(i)
Since, ray OD stands on line AB
` :. " "/_AOC= /_AOD =180^(@) " "` [linear pair axiom].....(ii)

From Eqs. (i) and (ii),
`/_AOC +/_AOD =/_AOD +/_`BOD
`rArr " "/_AOC= /_`BOD
(ii) Since, ray OD stands on line AB
`:." "/_AOD + /_BOD =180^(@) " " `[linear pair axiom].....(iii)
Since. ray OB stand on line CD.
`:." "/_DOB +/_BOD =180^(@) " " ....(iv)`
From Eqs. (iii) and (iv),
`/_AOD =/_BOD = /_DOB + /_ BOC`
`rArr " "/_AOD +/_BOC " "` Hence proved.