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Name the edge of these two half-planes           Find ( s2 ∪ k )'           Find s1 ∪ s2 ∪ k

The image shows a geometric representation involving two half-planes \( s_1 \) and \( s_2 \), and a line \( k \) that seems to act as a boundary between these half-planes. Let's go through the steps to address the questions:

1. **Naming the Edge of the Two Half-Planes:**
   - The line \( k \) is the edge or boundary between the half-planes \( s_1 \) and \( s_2 \). In this context, \( k \) is likely the line dividing \( s_1 \) and \( s_2 \).

2. **Finding \( (s_2 \cup k)' \):**
   - \( s_2 \cup k \) represents the union of the half-plane \( s_2 \) and the line \( k \).
   - The complement \( (s_2 \cup k)' \) would then represent everything outside \( s_2 \) and \( k \), which means it includes the half-plane \( s_1 \) that does not overlap with \( s_2 \cup k \).

3. **Finding \( s_1 \cup s_2 \cup k \):**
   - This represents the union of the half-plane \( s_1 \), the half-plane \( s_2 \), and the line \( k \).
   - Essentially, this union should cover the entire plane because \( s_1 \) and \( s_2 \) are divided by \( k \), and their union along with \( k \) will encompass the entire area.

To summarize:
- The edge of the two half-planes is the line \( k \).
- \( (s_2 \cup k)' \) is everything outside \( s_2 \) and \( k \), likely corresponding to \( s_1 \).
- \( s_1 \cup s_2 \cup k \) is the entire plane.

Would you like more details or have any questions about these findings?

### Relative Questions to Expand Understanding:
1. What is the significance of a half-plane in geometry?
2. How do unions and complements operate in set theory and geometry?
3. Why is the line \( k \) important in defining half-planes?
4. Can \( s_1 \) and \( s_2 \) overlap in this context?
5. How would the results change if \( k \) was not included in the union of \( s_1 \) and \( s_2 \)?

### Tip:
When dealing with sets and their operations (union, intersection, complement), it's helpful to visualize them on a diagram to better understand their spatial relationships.

Name the edge of these two half-planes. Find (s2 ∪ k)'. Find s1 ∪ s2 ∪ k.

Explore a geometric problem involving half-planes and set operations. Learn how to find the union and complement of sets with detailed step-by-step explanations. This problem is suitable for students in Grades 9-12 studying geometry and set theory.

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