1. **Problem Statement:** Given two parallel lines $l \parallel n$ cut by a transversal, with angles labeled $68^\circ$, $x^\circ$, and $(4y - 52)^\circ$, find the values of $x$ and $y$. 2. **Key Angle Relationships:** When a transversal crosses parallel lines: - Corresponding angles are equal. - Alternate interior angles are equal. - Adjacent angles on a straight line sum to $180^\circ$. 3. **Identify Angles:** The $68^\circ$ angle and $x^\circ$ are corresponding angles, so: $$x = 68$$ 4. **Use Adjacent Angles:** The angles $x^\circ$ and $(4y - 52)^\circ$ are adjacent and form a straight line, so their sum is $180^\circ$: $$x + (4y - 52) = 180$$ Substitute $x = 68$: $$68 + 4y - 52 = 180$$ 5. **Simplify and Solve for $y$:** $$\cancel{68} + 4y - \cancel{52} = 180$$ $$16 + 4y = 180$$ Subtract 16 from both sides: $$4y = 180 - 16$$ $$4y = 164$$ Divide both sides by 4: $$\frac{4y}{\cancel{4}} = \frac{164}{\cancel{4}}$$ $$y = 41$$ 6. **Final Answers:** $$x = 68$$ $$y = 41$$