The given diagram shows intersecting lines, and we need to find the values of the missing angles $d$, $e$, and $f$. There are a few angle relationships we can use here:

1. Vertically opposite angles: Vertically opposite angles are equal when two lines intersect.
2. Linear pair: Angles that form a straight line add up to $180^\circ$.

Step-by-step solution:

• Angle $d$:

  • Angle $d$ and $20^\circ$ form a linear pair (since they are on a straight line), so: $d + 20^\circ = 180^\circ$ Solving for $d$: $d = 180^\circ - 20^\circ = 160^\circ$

• Angle $e$:

  • Angle $e$ is vertically opposite to $121^\circ$, so: $e = 121^\circ$

• Angle $f$:

  • Angle $f$ and $121^\circ$ form a linear pair (since they are on a straight line), so: $f + 121^\circ = 180^\circ$ Solving for $f$: $f = 180^\circ - 121^\circ = 59^\circ$

Final answers:

• $d = 160^\circ$
• $e = 121^\circ$
• $f = 59^\circ$

Would you like any additional details on this? Here are five related questions for further exploration:

1. How are vertically opposite angles used to solve angle problems in more complex diagrams?
2. What are supplementary angles, and how do they apply to linear pairs?
3. How can you use angle bisectors to solve for unknown angles in intersecting lines?
4. How does the sum of angles around a point relate to these types of problems?
5. What strategies can help simplify solving multiple unknown angles in geometry?

Tip: Always look for linear pairs and vertically opposite angles in intersecting line diagrams to quickly solve for unknowns.