1. **Problem statement:** Find the angles $a$, $b$, and $c$ marked in the circle with diameter $AB$ and chords $AC$, $DB$, $AD$, and $BC$ intersecting inside the circle. 2. **Given:** $\angle BAC = 18^\circ$, an angle near the intersection toward $C$ is $50^\circ$, and we need to find $a = \angle ABD$, $b = \angle DAC$, and $c = \angle CBD$. 3. **Key properties:** - $AB$ is a diameter, so $\angle ACB = 90^\circ$ (angle in a semicircle). - Opposite angles formed by intersecting chords are equal. - The sum of angles around a point is $360^\circ$. 4. **Step to find $b$:** Since $b = \angle DAC$ and $\angle BAC = 18^\circ$, and $b$ lies between $AD$ and $AC$, we use the fact that $\angle BAC = b + 18^\circ$ or $b = 18^\circ$ (given directly as the angle at $A$ between $AD$ and $AC$ is $b$). 5. **Step to find $a$ and $c$:** Chords $AC$ and $DB$ intersect inside the circle. The angle formed between these chords at the intersection is $50^\circ$. By the intersecting chords theorem, the angle between chords is half the sum of the arcs intercepted. 6. **Using the intersecting chords theorem:** $$\angle = \frac{1}{2}(\text{arc } AD + \text{arc } BC) = 50^\circ$$ 7. **Since $a$ and $c$ are angles at $B$ between $BA$ and $BD$, and $BC$ and $BD$ respectively, and $AB$ is diameter, we use the fact that $a + c = 90^\circ$ (angles subtended by diameter). 8. **From the geometry and given angles, we find:** $$a = 32^\circ, \quad b = 18^\circ, \quad c = 58^\circ$$ **Final answers:** $$a = 32^\circ, \quad b = 18^\circ, \quad c = 58^\circ$$