O is a point on AB . ∠ AOC = 60 ° and OD → bisects ∠ COB . Find ∠ BOD . Does bisect ∠ AOD ?

From the figure, we have ∠ AOC = 60 ° , ∠ COD = ∠ BOD (since OD → bisects ∠ COB ) To find ∠ BOD . ∠ AOC and ∠ BOC are linear pair of angles, they are supplementary. So, ∠ AOC + ∠ BOC = 180 ° ⇒ 60 ° + ∠ BOC = 180 ° ⇒ ∠ BOC = 120 ° ⇒ ∠ COD + ∠ BOD = 120 ° ⇒ 2 ∠ BOD = 120 ° ∵ ∠ COD = ∠ BOD ⇒ ∠ BOD = 60 ° ⇒ ∠ COD = 60 ° Since ∠ AOD = ∠ AOC + ∠ COD and ∠ AOC = ∠ COD = 60 ° , ∠ AOD is bisected by OC . Hence, ∠ BOD = 60 ° and ∠ AOD is bisected by OC .

OC ¯ is perpendicular to AB ¯ at the point O . OE ¯ bisects ∠ COB and OF ¯ bisects ∠ AOC . Find ∠ FOE . Can you say ∠ AOF and ∠ BOE are congruent?

Given, OC ¯ is perpendicular to AB ¯ . Therefore, ∠ AOC = ∠ BOC = 90 ° . Also given, OE ¯ bisects ∠ COB . Therefore, ∠ BOE = ∠ COE = 45 ° OF ¯ bisects ∠ AOC . Therefore, ∠ AOF = ∠ COF = 45 ° Now, from the figure. ∠ FOE = ∠ COF + ∠ COE ⇒ ∠ FOE = 45 ° + 45 ° ⇒ ∠ FOE = 90 ° We know that, from the above calculation. ∠ AOF = ∠ BOE = 45 ° . Here, both the angles are equal. Therefore, ∠ AOF , ∠ BOE are congruent.