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To solve for the missing angles \(d\), \(e\), and \(f\) in the given diagram, we can use the following properties of angles:

1. **Linear pair**: Two angles that form a straight line sum up to \(180^\circ\).
2. **Vertically opposite angles**: When two lines intersect, opposite angles are equal.

### Steps:

1. **Angle \(d\)**:
   - \(d\) and the \(36^\circ\) angle are vertically opposite. So, \(d = 36^\circ\).

2. **Angle \(e\)**:
   - \(e\) and the \(102^\circ\) angle form a linear pair, so:
   \[
   e + 102^\circ = 180^\circ
   \]
   Solving for \(e\):
   \[
   e = 180^\circ - 102^\circ = 78^\circ
   \]

3. **Angle \(f\)**:
   - \(f\) and \(e\) are vertically opposite angles, so:
   \[
   f = e = 78^\circ
   \]

### Final answers:
- \(d = 36^\circ\)
- \(e = 78^\circ\)
- \(f = 78^\circ\)

Would you like further clarification or more details?

Here are 5 additional related questions:
1. How do complementary and supplementary angles differ?
2. How would the solution change if the angles given were different?
3. What are alternate interior and exterior angles?
4. Can you explain the difference between adjacent and vertical angles?
5. How can the concept of transversals be applied in geometry problems like this?

**Tip:** Always double-check angle sums, especially when working with linear pairs and intersecting lines!

Learn how to find missing angles using the properties of linear pairs and vertically opposite angles. In this geometry problem, we solve for angles d, e, and f, using a diagram and step-by-step solutions.

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