To find the values of $x$, $y$, and $z$, let's analyze the angles given in the diagram and apply the properties of parallel lines and angle sums in triangles.

Given:

• $\angle P = x$
• $\angle Q = y$
• $\angle R = z$
• $AB \parallel CD$
• $EF \parallel GH$
• $\angle F = 60^\circ$

Step-by-Step Solution:

1. Identify Relationships:

   • Since $AB \parallel CD$ and are intersected by transversal $EF$, the corresponding angles on these lines are equal.
   • Similarly, $EF \parallel GH$ implies corresponding angles on these lines are equal.

2. Calculate $x$:

   • Given that $\angle F = 60^\circ$, and since $EF \parallel GH$, the corresponding angle at point $P$ (denoted as $x$) will also be $60^\circ$.
   • Thus, $x = 60^\circ$.

3. Calculate $y$:

   • Using the alternate interior angles property with $AB \parallel CD$ and transversal $GH$, $\angle Q$ (denoted as $y$) is also equal to the angle given at $P$ because of the corresponding angles.
   • Therefore, $y = 60^\circ$.

4. Calculate $z$:

   • Given the parallel lines and the transversal properties, $\angle R$ (denoted as $z$) can be found using the sum of angles in the intersecting lines and the properties of parallel lines.
   • Since $x$ and $y$ are $60^\circ$, $z$ is found by considering the supplementary angles property.
   • Thus, $z = 180^\circ - (x + y) = 180^\circ - (60^\circ + 60^\circ) = 60^\circ$.

Summary:

• $x = 60^\circ$
• $y = 60^\circ$
• $z = 60^\circ$

Would you like any further details or have any questions about this solution?

Here are some related questions to consider:

1. How do you prove the properties of parallel lines with a transversal?
2. What are corresponding angles and how are they identified?
3. How can the sum of angles in a triangle be used in various geometric problems?
4. What are alternate interior angles and how do they relate to parallel lines?
5. Can you explain the concept of supplementary angles?
6. How do you identify and use angle relationships in intersecting lines?
7. What are the different types of angles formed when two parallel lines are intersected by a transversal?
8. How does understanding angle properties help in solving real-world geometry problems?

Tip: Always look for parallel lines and transversal relationships in geometric diagrams to simplify angle calculations.