1. The problem states that we have two pairs of intersecting lines creating vertical angles, with one angle given as $124^\circ$ and angles $a$ and $b$ marked at the intersections. 2. Vertical angles are equal when two lines intersect. Therefore, angle $a$ is vertically opposite to the $124^\circ$ angle. 3. By the vertical angle theorem, we have: $$a = 124^\circ$$ 4. Angle $b$ is adjacent to angle $124^\circ$ and forms a linear pair with it, meaning their sum is $180^\circ$ because they lie on a straight line. 5. Using the linear pair property: $$b + 124^\circ = 180^\circ$$ 6. Solving for $b$: $$b = 180^\circ - 124^\circ$$ $$b = 56^\circ$$ 7. Final answers: $$a = 124^\circ$$ $$b = 56^\circ$$