1. **Problem statement:** Find the values of the angles marked with letters in the given diagrams. 2. **Focus on part (b):** Given $b=80^\circ$ and $a=\frac{3}{2}c$, find angles $a$ and $c$ in triangle ABC. 3. **Recall the triangle angle sum rule:** The sum of interior angles in a triangle is $180^\circ$. 4. **Write the equation:** $$a + b + c = 180^\circ$$ Substitute $b=80^\circ$ and $a=\frac{3}{2}c$: $$\frac{3}{2}c + 80 + c = 180$$ 5. **Combine like terms:** $$\frac{3}{2}c + c = \frac{3}{2}c + \frac{2}{2}c = \frac{5}{2}c$$ So, $$\frac{5}{2}c + 80 = 180$$ 6. **Solve for $c$:** $$\frac{5}{2}c = 180 - 80 = 100$$ Multiply both sides by $\frac{2}{5}$: $$c = 100 \times \frac{2}{5} = 40^\circ$$ 7. **Find $a$ using $a=\frac{3}{2}c$:** $$a = \frac{3}{2} \times 40 = 60^\circ$$ 8. **Check sum:** $$a + b + c = 60 + 80 + 40 = 180^\circ$$ which confirms the solution. **Final answer:** $$a = 60^\circ, \quad c = 40^\circ$$