In the figure, O is the center of the circle, ∠AOE = 150°, ∠DAO = 51°. Calculate the sizes of the angles CEB and OCE.

We know that,

Angle at the center is double the angle at the circumference subtended by the same chord.

⇒ Reflex ∠AOE = 2∠ADE

⇒ ∠ADE = $\dfrac{1}{2}$ Reflex ∠AOE

⇒ ∠ADE = $\dfrac{1}{2}$ (360° - 150°)

⇒ ∠ADE = $\dfrac{1}{2} \times 210°$ = 105°.

From figure,

⇒ ∠DAB + ∠BED = 180° [Sum of opposite angles in a cyclic quadrilateral = 180°.]

⇒ ∠BED = 180° - ∠DAB = 180° - 51° = 129°.

Also,

⇒ ∠CEB + ∠BED = 180° [As CED is a straight line]

⇒ ∠CEB = 180° - ∠BED = 180° - 129° = 51°.

In △ADC,

⇒ ∠ADC + ∠ACD + ∠DAC = 180°

⇒ ∠ADE + ∠ACD + ∠DAO = 180° [From figure, ∠ADC = ∠ADE and ∠DAC = ∠DAO]

⇒ 105° + ∠ACD + 51° = 180°

⇒ ∠ACD = 180° - 105° - 51° = 24°.

From figure,

∠OCE = ∠ACD = 24°.

Hence, ∠CEB = 51° and ∠OCE = 24°.