https://file.aiquickdraw.com/mathbot/file/1726597678198fuuv474g.jpg

Let's go through each question step by step based on the diagram provided:

### 1. \( \angle AHC \) is adjacent to \( \angle AHF \) and _______.
- Since \( \angle AHC \) and \( \angle AHB \) are adjacent, the second answer is **\( \angle AHB \)** (already filled in by you).

### 2. \( \angle BHD \) is adjacent to \( \angle FHB \) and _______.
- \( \angle BHD \) is adjacent to \( \angle GHD \), so the second answer is **\( \angle GHD \)**.

### 3. \( \angle FHB \) and _______ are vertical angles.
- Vertical angles are opposite each other. Therefore, \( \angle FHB \) and \( \angle CHA \) are vertical angles. The answer is **\( \angle CHA \)**.

### 4. \( \angle FHC \) and _______ are vertical angles.
- \( \angle FHC \) is vertical to \( \angle BHD \), so the answer is **\( \angle BHD \)**.

### 5. \( \angle LFJ \) and _______ are supplementary angles.
- \( \angle LFJ \) and \( \angle JFH \) are supplementary because they form a straight line. The answer is **\( \angle JFH \)**.

### 6. \( \angle FHC \) and _______ are complementary angles.
- \( \angle FHC \) and \( \angle FHB \) are complementary because their sum is \( 90^\circ \) (as given in the problem). The answer is **\( \angle FHB \)**.

### 7. \( \angle JFH \) and _______ are supplementary angles.
- \( \angle JFH \) and \( \angle LFJ \) are supplementary as they form a straight line. The answer is **\( \angle LFJ \)**.

### 8. \( \angle BHD \) and _______ are complementary angles.
- Since \( \angle FHC \) and \( \angle BHD \) are complementary (from previous deductions), the answer is **\( \angle FHC \)**.

### 9. If \( m \angle CHA = 40^\circ \), then \( m \angle BHD = \) _______.
- \( \angle CHA \) and \( \angle BHD \) are vertical angles, so they are congruent. If \( m \angle CHA = 40^\circ \), then \( m \angle BHD = 40^\circ \). The answer is **40°**.

### Recap of answers:
1. \( \angle AHF \) and \( \angle AHB \)
2. \( \angle FHB \) and \( \angle GHD \)
3. \( \angle CHA \)
4. \( \angle BHD \)
5. \( \angle JFH \)
6. \( \angle FHB \)
7. \( \angle LFJ \)
8. \( \angle FHC \)
9. \( 40^\circ \)

Let me know if you need further explanations or details!

### Follow-up Questions:
1. What are the properties of vertical angles?
2. How do you determine if two angles are adjacent?
3. What is the difference between complementary and supplementary angles?
4. Can you explain why \( m \angle BHD = 40^\circ \) based on vertical angles?
5. How can you identify straight angles from the diagram?

**Tip:** Vertical angles are always equal, regardless of the size of the angles.

Solve 1 through 9 based on the geometry diagram involving adjacent, vertical, supplementary, and complementary angles.

Solve a geometry problem involving adjacent, vertical, supplementary, and complementary angles using a provided diagram. Includes detailed explanations for each angle type and their relationships. Suitable for students in Grades 6-8.

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