1. **Stating the problem:** We are given a figure with several intersecting lines and angles labeled as 80°, w°, 64°, x°, y°, z°, and 51°. We need to find the values of the unknown angles $w$, $x$, $y$, and $z$. 2. **Using angle rules:** At the central intersection, the angles around a point sum to 360°. Also, vertically opposite angles are equal, and angles on a straight line sum to 180°. Parallel lines crossed by a transversal create equal alternate interior angles. 3. **Find $w$:** At the central intersection, the angles are 80°, $w$, and 64°. Since these three angles plus the unknown fourth angle sum to 360°, and the fourth angle is vertically opposite to $w$, we have: $$80 + w + 64 + w = 360$$ $$144 + 2w = 360$$ $$2w = 360 - 144$$ $$2w = 216$$ $$w = \frac{216}{2} = 108$$ 4. **Find $x$:** The angle $x$ is on the rising diagonal crossing the upper parallel line. Since the upper and lower lines are parallel, and the diagonal is a transversal, $x$ is equal to the alternate interior angle to $w$, so: $$x = w = 108$$ 5. **Find $y$:** The angle $y$ is on the steep line crossing the lower parallel line. Since the lower parallel line is parallel to the upper one, and the steep line crosses it, $y$ is supplementary to $x$ because they form a linear pair: $$y + x = 180$$ $$y + 108 = 180$$ $$y = 180 - 108 = 72$$ 6. **Find $z$:** The angle $z$ is at the apex of a triangle with angles $z$, 51°, and $y$. The sum of angles in a triangle is 180°, so: $$z + 51 + y = 180$$ $$z + 51 + 72 = 180$$ $$z + 123 = 180$$ $$z = 180 - 123 = 57$$ **Final answers:** $$w = 108^\circ, x = 108^\circ, y = 72^\circ, z = 57^\circ$$