From the image, we can see that two pairs of parallel lines are given: $l \parallel m$ and $p \parallel q$. The task is to find the angles $a$, $b$, $c$, and $d$. Additionally, we are provided with an angle of $60^\circ$ in the diagram.

Step-by-step Solution:

1. Identify the transversal and relationships:

   • Lines $l$ and $m$ are parallel and are cut by the transversal line $p$.
   • Similarly, lines $p$ and $q$ are parallel and are cut by the transversal line $l$.
   • These parallel lines and the transversal create corresponding and alternate angles, which are congruent.

2. Given Information:

   • The angle adjacent to $a$ is $60^\circ$.
   • By the linear pair property, $a$ is supplementary to this angle: $a = 180^\circ - 60^\circ = 120^\circ$

3. Angle Relationships:

   • Since $l \parallel m$, angle $a$ and angle $b$ are corresponding angles. Therefore, $b = 120^\circ$.
   • $b$ and $d$ are vertically opposite angles, so $d = 120^\circ$.
   • Angle $c$ is adjacent to $d$, forming a linear pair. Therefore: $c = 180^\circ - 120^\circ = 60^\circ$

Final Answers:

• $a = 120^\circ$
• $b = 120^\circ$
• $c = 60^\circ$
• $d = 120^\circ$

Would you like more details on any of these steps or further clarification?

Here are five related questions to explore further:

1. What is the relationship between alternate interior angles when parallel lines are cut by a transversal?
2. How do corresponding angles help in solving problems involving parallel lines?
3. Can two vertically opposite angles ever be unequal?
4. How do you find angles when two lines intersect?
5. What role does the linear pair property play in geometry?

Tip: Always look for corresponding, alternate, and vertically opposite angles when working with parallel lines and transversals!