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.
Solution:
We will use the following concepts to answer the question:

A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle.

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

The sum of angles in a triangle is 180°.

Angles in the same segment are equal.
Based on the data given, let's draw the figure as shown below.
In the triangles, ABD and BCD, ∠CAD = ∠CBD = 70°. (Angles in the same segment are equal)
Hence, ∠BAD = ∠CAB + ∠DAC
= 30° + 70° = 100°
Thus, ∠BAD = 100°
Since ABCD is a cyclic quadrilateral, the sum of either pair of opposite angles of a cyclic quadrilateral is 180º.
∠BAD + ∠BCD = 180°
∠BCD = 180°  100°
= 80°
Thus, ∠BCD = 80°
Also given AB = BC.
So, ∠BCA = ∠BAC = 30° (Base angles of isosceles triangle are equal)
∠ECD = ∠BCD  ∠BCA
= 80°  30°
= 50°
Thus, ∠ECD = 50°
Video Solution:
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.
Maths NCERT Solutions Class 9  Chapter 10 Exercise 10.5 Question 6:
Summary:
If ABCD is a cyclic quadrilateral whose diagonals intersect at a point E, ∠DBC = 70°, ∠BAC = 30°, and AB = BC, then ∠BCD = 80° and ∠ECD = 50°.