In this problem, you are given two pairs of parallel lines:

• Line $b \parallel c$
• Line $d \parallel f$

The angles provided in the diagram are:

• An angle of $82^\circ$ between lines $b$ and $d$
• An angle of $76^\circ$ between lines $d$ and $f$

You are tasked with finding the values of $x$ and $y$, which represent the unknown angles at the intersections of these lines.

Step-by-Step Solution:

1. Using the properties of parallel lines and transversals:
   Since lines $b \parallel c$, and angle $82^\circ$ is given at their intersection with line $d$, the angle corresponding to $x$ (the one on the other side of this intersection) is supplementary to $82^\circ$, because they form a linear pair. Therefore: $x + 82^\circ = 180^\circ$ Solving for $x$: $x = 180^\circ - 82^\circ = 98^\circ$

2. Using the alternate interior angles:
   Since lines $d \parallel f$ and $y$ is an interior angle formed by the transversal line $e$, angle $y$ is equal to the angle $76^\circ$ given in the diagram (since alternate interior angles are equal when lines are parallel): $y = 76^\circ$

Final Answer:

• $x = 98^\circ$
• $y = 76^\circ$

Do you want more details on how to apply these properties?

Here are 5 related questions for practice:

1. What is the sum of angles in a triangle?
2. How are corresponding angles defined when lines are parallel?
3. What is the relationship between supplementary angles?
4. Can alternate exterior angles also be equal when lines are parallel?
5. What are consecutive interior angles, and how do they behave with parallel lines?

Tip: Always look for relationships between angles formed by parallel lines and a transversal to quickly identify angle measures.