In figure, ∠ADC = 130° and chord BC = chord BE. Find ∠CBE.

Given ∠ADC = 130° and chord BC = chord BE.

Consider the points A, B, C and D which form a cyclic quadrilateral.

We know that in a cyclic quadrilateral the opposite angles are supplementary.

In cyclic quadrilateral ADCB,

⇒∠ADC + ∠OBC = 180°

⇒ 130° + ∠OBC = 180°

⇒∠OBC = 180° - 130° = 50°

Consider ΔBOC and ΔBOE,

⇒ BC = BE [given]

⇒ OC = OE [radii of same circle]

⇒ OB = OB [common side]

By SSS congruence rule,

⇒ ΔBOC ≅ ΔBOE

By CPCT,

⇒∠OBC = ∠OBE = 50°

⇒∠CBE = ∠CBO + ∠EBO

= 50° + 50°

∴ ∠CBE = 100°