1. **Stating the problem:** We have a triangle ABC with a point D on BC such that AD is perpendicular to BC. The angle at A adjacent to this perpendicular is 40°. At vertex B, there are two angles of 10° each, and we need to find the angle $x$ at vertex C adjacent to BC. 2. **Understanding the setup:** Since AD is perpendicular to BC, angle $ADB$ and angle $ADC$ are right angles (90°). 3. **Using angle sum in triangle ABD:** Triangle ABD has angles 90° (at D), 10° (at B), and the remaining angle at A is $40°$. Check if these add up correctly: $$90° + 10° + 40° = 140°$$ which is more than 180°, so the 40° angle at A is adjacent to the perpendicular but not inside triangle ABD. 4. **Using angle sum in triangle ABC:** The sum of angles in triangle ABC is 180°: $$\angle A + \angle B + \angle C = 180°$$ 5. **Given angles:** - $\angle A = 40°$ - $\angle B$ is composed of two 10° angles, so total $20°$ - $\angle C = x°$ 6. **Calculate $x$:** $$40° + 20° + x = 180°$$ $$x = 180° - 60° = 120°$$ 7. **Conclusion:** The angle $x$ at vertex C is $120°$.