Question: What are the values of $x$, $y$, and $z$? Use theorems to justify each answer. 74° $x°$ 66° $y°$ $z°$ By the Choose answer 1 Angles Theorem and the Triangle Choose answer 2 Theorem, $x = $ Choose answer 3.

1. **State the problem:** We are given a diagram with two vertical parallel lines and two slanted lines crossing them, forming angles labeled $74^\circ$, $x^\circ$, $66^\circ$, $y^\circ$, and $z^\circ$. We need to find the values of $x$, $y$, and $z$ using angle theorems. 2. **Identify the theorems:** - Since the vertical lines are parallel, corresponding angles and alternate interior angles are equal. - The sum of angles in a triangle is $180^\circ$. 3. **Find $x$:** - The angle $74^\circ$ and angle $x$ are alternate interior angles formed by the transversal crossing the parallel lines. - Therefore, by the Alternate Interior Angles Theorem, $x = 74^\circ$. 4. **Find $y$:** - At the upper-right intersection, angles $66^\circ$, $y^\circ$, and $x^\circ$ form a triangle. - Sum of angles in a triangle is $180^\circ$: $$66 + y + x = 180$$ - Substitute $x = 74$: $$66 + y + 74 = 180$$ - Simplify: $$140 + y = 180$$ - Solve for $y$: $$y = 180 - 140 = 40$$ 5. **Find $z$:** - The angle $z$ and angle $y$ are corresponding angles formed by the transversal crossing the parallel lines. - By the Corresponding Angles Theorem, $z = y = 40^\circ$. **Final answers:** $$x = 74^\circ, \quad y = 40^\circ, \quad z = 40^\circ$$