1. **Problem statement:** We are given a circle ABCD with center O, where AB = AC and \(\angle DBC = 40^\circ\). We need to find \(\angle ABD\). 2. **Given information and properties:** - Since AB = AC, triangle ABC is isosceles with \(AB = AC\). - \(\angle DBC = 40^\circ\). - Angles in the same segment of a circle are equal. 3. **Using the property of angles in the same segment:** - \(\angle BDA = \angle DBC = 40^\circ\). 4. **Using the isosceles triangle property:** - In triangle ABD, since \(\angle BDA = 40^\circ\) and \(\angle ABD = \angle ADB\) (angles opposite equal sides), we have \(\angle ABD = 40^\circ\). 5. **Conclusion:** - Therefore, \(\boxed{\angle ABD = 40^\circ}\).