In the given figure, O is the centre of the circle. If the angles ∠AOB, ∠OAC are, respectively, 140° and 50°, then the value of angle ∠BAC is ______.

Given:

O is the center of the circle.

∠AOB = 140°

∠OAC = 50°

Formula Used:

Angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circle.

In a triangle, angles opposite to equal sides are equal.

Sum of angles in a triangle is 180°.

Calculation:

Consider triangle OAC. Since OA and OC are radii of the same circle, OA = OC.

Therefore, ∠OCA = ∠OAC = 50° (angles opposite to equal sides are equal).

In triangle OAC, ∠AOC + ∠OAC + ∠OCA = 180° (sum of angles in a triangle).

⇒ ∠AOC + 50° + 50° = 180°

⇒ ∠AOC + 100° = 180°

⇒ ∠AOC = 180° - 100°

⇒ ∠AOC = 80°

The reflex angle ∠AOC at the center is 360° - ∠AOC = 360° - 80° = 280°.

Angle subtended by arc ABC at the center is reflex ∠AOC = 280°.

Angle subtended by arc ABC at the circumference is ∠ABC.

∠ABC = (1/2) × reflex ∠AOC = (1/2) × 280° = 140°.

Consider triangle OAB. Since OA and OB are radii of the same circle, OA = OB.

Therefore, ∠OBA = ∠OAB.

In triangle OAB, ∠AOB + ∠OAB + ∠OBA = 180°.

⇒ 140° + ∠OAB + ∠OAB = 180°

⇒ 140° + 2∠OAB = 180°

⇒ 2∠OAB = 180° - 140°

⇒ 2∠OAB = 40°

⇒ ∠OAB = 20°

∠BAC = ∠OAC - ∠OAB = 50° - 20° = 30°.

∴ The value of angle ∠BAC is 30°.