1. **Problem statement:** You asked how we found that $\angle BDA = 40^\circ$ given $\angle DBC = 40^\circ$ in the circle. 2. **Key property used:** Angles subtended by the same chord in the same segment of a circle are equal. 3. **Explanation:** - The chord here is $BD$. - $\angle DBC$ and $\angle BDA$ both subtend chord $BD$ but lie in the same segment of the circle. - Therefore, by the circle theorem, $\angle BDA = \angle DBC = 40^\circ$. 4. **Summary:** This is a fundamental property of circle geometry that angles in the same segment are equal, which allowed us to conclude $\angle BDA = 40^\circ$.