Question

In the following figure, ‘O’ is the centre of the circle and ∠AOC = 110°. Find ∠ABC.

Answer 1

Given:

O is the centre of the circle.

∠AOC = 110°.

∠ABC

Step 1:

Use the property of angles subtended by the same chord. The angle at the center (∠AOC) is twice the angle at the circumference subtended by the same chord AC.

\[\angle ABC = \frac{1}{2} \times \angle AOC = \frac{1}{2} \times 110^{\circ} = 55^{\circ}\]

But in the cyclic quadrilateral ABCD (considering points A, B, C, and the point diametrically opposite B or using supplementary angles in cyclic quadrilateral):

Opposite angles of a cyclic quadrilateral sum to 180°.

The relevant opposite angle to ∠ABC is ∠ADC, but since B lies on the circle, and angles ∠ABC and ∠ADC are opposite angles of the cyclic quadrilateral, the following relation applies:

∠ABC + ∠ADC = 180°

From Step 1, ∠ADC = 55° (angle subtended at circumference by chord AC opposite to ∠AOC)

∠ABC + 55° = 180°

∠ABC = 180° −  55°

= 125°