Question

In the given figure, if ∠ADC = 130^∘ and chord BC = chord BE, then ∠CBE = _______.

Answer 1

Given:
∠ADC = 130^∘ ...(1)
chord BC = chord BE ...(2)


Quadrilateral ADCB is a cyclic quadrilateral.

In a cyclic quadrilateral, the sum of opposite angles is 180^∘.
Thus, ∠ADC + ∠CBA = 180°
⇒ 130° + ∠CBA = 180°
⇒ ∠CBA = 180° − 130°
⇒ ∠CBA = 50° ...(3)


In ∆CBO and ∆EBO,
BC = BE (given)
OB = OB (common)
OC = OE (radius of the circle)

By SSS property,
∆OCB ≅ ∆OEB

Therefore, ∠OBC = ∠OBE = 50° (by C.P.C.T.) ...(4)

Thus, ∠CBE = ∠OBE + ∠OBC
= 50° + 50° (From (4))
= 100°


Hence, ∠CBE = 100°.