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The problem asks to find the measures of missing angles \(d\), \(e\), and \(f\) in the diagram, given that one angle is \(112^\circ\) and another is \(42^\circ\). These angles form a pair of intersecting lines, creating a system of vertical and adjacent angles.

### Step-by-Step Solution:

1. **Vertical Angles**: 
   - Angle \(d\) is vertical to the \(42^\circ\) angle, and vertical angles are equal. Therefore, 
   \[
   d = 42^\circ.
   \]

2. **Supplementary Angles**: 
   - Angle \(e\) and the \(112^\circ\) angle are on a straight line, meaning they are supplementary. The sum of supplementary angles is \(180^\circ\). Thus,
   \[
   e = 180^\circ - 112^\circ = 68^\circ.
   \]

3. **Vertical Angles Again**:
   - Angle \(f\) is vertical to angle \(e\), and again, vertical angles are equal. Therefore,
   \[
   f = 68^\circ.
   \]

### Final Answer:
- \(d = 42^\circ\)
- \(e = 68^\circ\)
- \(f = 68^\circ\)

Would you like more details or explanations on any specific step?

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Here are some related questions you could explore:
1. What are complementary angles, and how are they different from supplementary angles?
2. How would the problem change if two of the angles were complementary?
3. What other types of angle pairs exist besides vertical and supplementary?
4. How do we use adjacent angles to solve for unknowns?
5. What are the real-world applications of angle properties?

**Tip**: Always remember that vertical angles are equal and supplementary angles add up to \(180^\circ\). This simplifies many angle problems!

Find the measure of the missing angles in a diagram where two angles are given: 112° and 42°, forming a system of intersecting lines.

This problem asks to find the missing angles (d, e, f) in a diagram involving vertical and supplementary angles. Given angles are 112° and 42°. Solve using the Vertical Angles Theorem and the Supplementary Angles Theorem.

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