To solve for angles $d$, $e$, and $f$, let's analyze the diagram provided.

1. Identify the angles:
   • $d$ and $121^\circ$ are vertical angles, which means $d = 121^\circ$.
   • $e$ is a corresponding angle to $121^\circ$, meaning $e = 121^\circ$.
   • $f$ is a corresponding angle to $57^\circ$, so $f = 57^\circ$.

Thus, the values are:

• $d = 121^\circ$
• $e = 121^\circ$
• $f = 57^\circ$

Would you like further details on this, or do you have any questions?

Here are some related questions you might find interesting:

1. What are vertical angles, and why are they equal?
2. How do you identify corresponding angles in parallel lines?
3. How can you use the properties of parallel lines to find unknown angles?
4. What is the relationship between supplementary angles?
5. How does alternate interior angles theorem apply in parallel line situations?

Tip: When working with angles formed by parallel lines and a transversal, always look for pairs of vertical, corresponding, and alternate interior angles—they are often equal or supplementary.