1. **Problem statement:** Triangle ABC is isosceles with AC parallel to BD. Given angle C is 40°, find the values of angles $a$ and $b$. 2. **Key facts and formulas:** - In an isosceles triangle, two sides are equal, so the angles opposite those sides are equal. - Parallel lines imply alternate interior angles are equal. - The sum of angles in a triangle is $180^\circ$. 3. **Step-by-step solution:** 1. Since AC is parallel to BD, angle $a$ at vertex A is equal to angle C (alternate interior angles), so $a = 40^\circ$. 2. Triangle ABC is isosceles with AC = BC, so angles opposite these sides are equal. Angle C is $40^\circ$, so angle B is also $40^\circ$. 3. Sum of angles in triangle ABC: $$a + b + 40^\circ = 180^\circ$$ Substitute $a = 40^\circ$: $$40^\circ + b + 40^\circ = 180^\circ$$ 4. Simplify: $$b + 80^\circ = 180^\circ$$ 5. Solve for $b$: $$b = 180^\circ - 80^\circ = 100^\circ$$ **Final answers:** $$a = 40^\circ$$ $$b = 100^\circ$$