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In this problem, we are asked to find the measures of the missing angles: \(d\), \(e\), and \(f\). Let's analyze the figure step by step:

1. **Angle \(d\)**: We see that angle \(d\) forms a right angle (90°) with the horizontal and vertical lines. Therefore:
   \[
   d = 90^\circ
   \]

2. **Angle \(f\)**: Angle \(f\) is vertically opposite to the given 47° angle. Vertically opposite angles are equal. Thus:
   \[
   f = 47^\circ
   \]

3. **Angle \(e\)**: Angle \(e\) and the 47° angle are supplementary angles, as they form a straight line along the vertical. Supplementary angles add up to 180°. Therefore:
   \[
   e = 180^\circ - 47^\circ = 133^\circ
   \]

Thus, the missing angles are:
\[
d = 90^\circ, \quad e = 133^\circ, \quad f = 47^\circ
\]

Would you like any further explanation or details?

Here are five related questions to extend your understanding:
1. What are vertically opposite angles, and why are they always equal?
2. How do we determine whether two angles are supplementary?
3. Can you find other examples where two angles are vertically opposite?
4. What is the sum of all interior angles at any intersection of two straight lines?
5. How would the situation change if the 47° angle were changed to another value?

**Tip:** Always remember that vertically opposite angles are equal and adjacent angles on a straight line are supplementary. This can help simplify many geometric problems!

Find the measure of the missing angles d, e, and f.

In this geometry problem, find the missing angles d, e, and f using the properties of vertically opposite angles and supplementary angles. Step-by-step solutions included for each missing angle. Suitable for students in Grades 8-10.

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