ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD.

Cyclic quadrilateral ABCD is shown in the figure below:

∠CAD = ∠CBD = 70° (Angles in the same segment are equal)

From figure,

⇒ ∠BAD = ∠CAB + ∠DAC

⇒ ∠BAD = 30° + 70°

⇒ ∠BAD = 100°.

Since ABCD is a cyclic quadrilateral, the sum of opposite angles of a cyclic quadrilateral is 180°.

⇒ ∠BAD + ∠BCD = 180°

⇒ ∠BCD = 180° - ∠BAD

⇒ ∠BCD = 180° - 100° = 80°

Given, AB = BC

In △ ABC,

⇒ ∠BCA = ∠BAC = 30° (Angles opposite to equal sides are equal)

From figure,

⇒ ∠ECD = ∠BCD - ∠BCA

⇒ ∠ECD = 80° - 30° = 50°.

Hence, ∠BCD = 80° and ∠ECD = 50°.