1. **Problem Statement:** Given that lines $AB \parallel CD$ and $CD \parallel EF$, prove a relationship involving the angles formed by a transversal crossing these parallel lines. 2. **Understanding the Problem:** When a transversal crosses parallel lines, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary. 3. **Given:** - $AB \parallel CD$ - $CD \parallel EF$ 4. **To Prove:** Since $AB \parallel CD$ and $CD \parallel EF$, by the transitive property of parallel lines, $AB \parallel EF$. 5. **Angle Relationships:** - Because $AB \parallel CD$, angle 1 ($x$) is equal to angle 2 ($y$) as corresponding angles. - Because $CD \parallel EF$, angle 3 is equal to angle 4 ($z$) as corresponding angles. 6. **Since angle 2 and angle 3 are vertically opposite angles, they are equal:** $$y = \text{angle 2} = \text{angle 3}$$ 7. **Therefore, by substitution:** $$x = y = z$$ 8. **Conclusion:** The angles $x$, $y$, and $z$ are equal due to the parallelism of the lines and the properties of corresponding and vertically opposite angles. This proves the required angle relationships based on the given parallel lines and transversal.