A, B and C are three points on a circle such that the angles subtended by the chords AB and AC at the centre O are $$80^\circ$$ and $$120^\circ$$, respectively. The value of $$\angle BAC $$ is:

In the $$\triangle OAB,

OB = OA(radius) so,

$$\angle OBA =  \angle BAO$$

$$\angle OBA + \angle BAO + \angle AOB = 180\degree$$

$$\angle BAO + \angle BAO + 80\degree = 180\degree$$

$$2\angle BAO = 180 - 80 = 100\degree$$

$$\angle BAO = 50\degree$$

In the $$\triangle OAC,

OC = OA(radius) so,

$$\angle OAC = \angle OCA$$

$$\angle OAC + \angle OCA + \angle AOC = 180\degree$$

$$\angle OAC + \angleOAC + 120\degree = 180\degree$$

$$2\angle OAC = 180 - 120 = 60\degree$$

$$\angle OAC = 30\degree$$

$$\angle BAC = \angle BAO + \angle OAC$$

$$\angle BAC = 50 + 30 = 80\degree$$