Let's analyze the given diagram to find the missing angles.

Key observations:

1. The angle given between the two diagonal lines is $51^\circ$.
2. There is a right angle $d$ at the bottom formed by two perpendicular lines.
3. The lines intersect, forming a vertical angle relationship, which means angles directly opposite to each other are congruent.

Steps to find the missing angles:

1. Angle $e$:

   • $e$ and $51^\circ$ form a linear pair, meaning they are supplementary (sum to $180^\circ$).
   • So, $e = 180^\circ - 51^\circ = 129^\circ$.

2. Angle $f$:

   • Angle $f$ is vertically opposite to $51^\circ$, so it is also $51^\circ$ (since vertical angles are congruent).

3. Angle $d$:

   • $d$ is a right angle, already marked as $90^\circ$, so no further calculation is needed.

Final values:

• Angle $e = 129^\circ$
• Angle $f = 51^\circ$
• Angle $d = 90^\circ$

Let me know if you would like more details or explanations.

Here are 5 related questions:

1. What are vertical angles, and how do they relate to congruence?
2. Can you explain what supplementary angles are and give more examples?
3. What is the sum of angles on a straight line, and why does this rule hold?
4. How would you calculate angles if the diagram involved parallel lines?
5. What are complementary angles, and how are they different from supplementary angles?

Tip: When solving geometry problems involving intersecting lines, always check for vertical and supplementary angles. They often provide quick solutions!