1. **Stating the problem:** We have a figure with two pairs of parallel lines intersected by a diagonal line, creating several angles labeled $z^\circ$, $y^\circ$, $44^\circ$, $x^\circ$, $55^\circ$, and $w^\circ$. We need to find relationships between these angles. 2. **Important rules:** - When two parallel lines are cut by a transversal, alternate interior angles are equal. - Corresponding angles are equal. - The sum of angles around a point is $360^\circ$. - The sum of angles on a straight line is $180^\circ$. 3. **Using the given angles:** - At the bottom-left intersection, angles $x^\circ$ and $55^\circ$ are adjacent and form a straight line, so: $$x + 55 = 180$$ 4. **Solving for $x$:** $$x = 180 - 55 = 125$$ 5. **At the central intersection, the angle is $44^\circ$. Since the diagonal crosses parallel lines, alternate interior angles are equal, so $w = 44$. 6. **At the top intersections, $z$ and $y$ are angles formed by the parallel lines and the diagonal. Using corresponding angles and linear pairs, we find: - $z = 55$ (corresponding to the $55^\circ$ angle at the bottom) - $y = x = 125$ (alternate interior angles) **Final answers:** $$x = 125^\circ, w = 44^\circ, z = 55^\circ, y = 125^\circ$$