To solve this problem, we need to find the missing angles in the diagram, using properties of angles.

Analysis:

1. Angle $78^\circ$ is given between lines $e$ and $f$.
2. $d$ is marked as a right angle, which is $90^\circ$.

Steps:

• Using vertical angles: Vertical angles are congruent, so the angle directly opposite $78^\circ$ (on the left side of the vertical line) will also be $78^\circ$.

• Using supplementary angles: When two angles form a straight line, they are supplementary, meaning their sum is $180^\circ$.

  • The angle adjacent to $78^\circ$ on line $f$ (let's call it $e$) will be $180^\circ - 78^\circ = 102^\circ$.

• Using right angle: Since $d$ is a right angle, the angle adjacent to it on line $e$ (on the top) will also be $90^\circ$.

Conclusion:

• The angles in the diagram are:
  • Vertical angle to $78^\circ$ = $78^\circ$
  • Angle adjacent to $78^\circ$ on line $f$ = $102^\circ$
  • Right angle $d = 90^\circ$
  • Angle adjacent to $d$ on line $e$ = $90^\circ$

Would you like further details on any specific step?

Here are five questions to expand on the information:

1. How do vertical angles help in finding unknown angles in geometry?
2. What is the relationship between supplementary angles and straight lines?
3. How can we apply these angle properties to more complex diagrams?
4. What is the difference between complementary and supplementary angles?
5. Why are vertical angles always congruent?

Tip: Always look for vertical and supplementary angles when you see intersecting lines. This helps in solving for unknown angles quickly.