In the given figure, AB is a side of a regular six-sided polygon and AC is a side of a regular eight-sided polygon inscribed in the circle with centre O. Calculate the sizes of :

(i) ∠AOB,

(ii) ∠ACB,

(iii) ∠ABC.

Join OC.

(i) We know that,

Each side of a regular hexagon, inscribed in a circle subtends an angle of 60° at the centre.

⇒ ∠AOB = 60°.

Hence, ∠AOB = 60°.

(ii) We know that,

Angle at the centre is twice the angle at remaining circumference.

∴ ∠AOB = 2∠ACB

⇒ ∠ACB = $\dfrac{1}{2}$∠AOB = $\dfrac{1}{2} \times 60°$ = 30°.

Hence, ∠ACB = 30°.

(iii) Since AC is the side of a regular octagon,

∠AOC = $\dfrac{360°}{8}$ = 45°.

Again, arc AC subtends ∠AOC at the centre and ∠ABC at the remaining part of the circle.

∴ ∠AOC = 2∠ABC

∴ ∠ABC = $\dfrac{1}{2}$ x ∠AOC = $\dfrac{1}{2}$ x 45° = 22.5°

Hence, ∠ABC = 22.5°.