Given is a circle with centre at C. A, B and D are the points on the circumference. Find ∠ABC.

Solution:

• Given: ∠BCD = 126° (central angle)
• Also, the angle at D between CD and DA is 33°, so in triangle ACD (where CA = CD), the base angles are equal.

Step 1: Find central angle ∠ACD

In triangle ACD:

• Base angles at A and D = 33° each
• So, central angle ∠ACD = 180° − (33° + 33°) = 114°

Step 2: Use angles around center C

Sum of central angles around point C:
∠ACB + ∠BCD + ∠DCA = 360°

∠ACB + 126° + 114° = 360°
∠ACB = 360° − 240° = 120°

Step 3: Find ∠ABC

Triangle ABC is isosceles (CA = CB), so base angles are equal.

Let each base angle = x
Then:
x + x + 120° = 180°
2x = 60°
x = 30°

Final Answer: $30^\circ$