In the given circle, O is the center. If \(\angle\) AOC = 100 degrees, find \(\angle\) ABC.

Solution:

The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

The angle subtended by the arc AC at the center is \( \angle AOC \). The angle subtended by the arc AC at the circumference is \( \angle ABC \).

Therefore, \( \angle AOC = 2 \times \angle ABC \).

Given \( \angle AOC = 100^{\circ} \).

So, \( 100^{\circ} = 2 \times \angle ABC \).

To find \( \angle ABC \), we divide \( \angle AOC \) by 2:

\[ \angle ABC = \frac{\angle AOC}{2} = \frac{100^{\circ}}{2} = 50^{\circ} \]

Ответ: \( \angle ABC = 50^{\circ} \).