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The image contains questions about angle relationships, specifically referring to vertical angles, linear pairs, and evaluating angle measures. Let's work through them one by one:

### Given Diagram Overview:
The angles are formed by two intersecting lines, and the numbers (1, 2, 3, 4) represent different angles.

### 9) **Give two vertical angles. What can you claim about these vertical angles?**

- **Vertical angles** are the angles opposite each other when two lines intersect. In the diagram:
  - Angle 1 (∠1) and Angle 3 (∠3) are vertical angles.
  - Angle 2 (∠2) and Angle 4 (∠4) are also vertical angles.
  
  **Claim**: Vertical angles are congruent, meaning their measures are equal. So, \( m∠1 = m∠3 \) and \( m∠2 = m∠4 \).

### 10) **Name two angles that form a linear pair. What can you claim about these two angles?**

- **Linear pairs** are adjacent angles that form a straight line, meaning their sum is 180° (supplementary angles). For example:
  - Angle 1 (∠1) and Angle 2 (∠2) form a linear pair.
  - Angle 3 (∠3) and Angle 4 (∠4) also form a linear pair.
  
  **Claim**: The sum of the angles in a linear pair is 180°. So, \( m∠1 + m∠2 = 180° \) and \( m∠3 + m∠4 = 180° \).

### 11) **Evaluate the following:**
   - **a. \( m∠1 + m∠4 \):** 
     Since ∠1 and ∠4 are not related by vertical or linear pair rules, their sum is unknown.
   - **b. \( m∠4 + m∠3 \):**
     This is a linear pair, so \( m∠4 + m∠3 = 180° \).
   - **c. \( m∠1 + m∠2 + m∠3 + m∠4 \):**
     These four angles form a full circle around the intersection of the two lines, so their sum is \( 360° \).

### 12) **"True" or "False." Explain your answer:**
   - **a. \( ∠1 ≅ ∠2 \):**
     False. ∠1 and ∠2 are a linear pair, and their sum is 180°, but they are not congruent.
   - **b. \( ∠1 ≅ ∠3 \):**
     True. ∠1 and ∠3 are vertical angles, and vertical angles are congruent.
   - **c. \( m∠1 = m∠3 \):**
     True. This is another way of stating that ∠1 and ∠3 are congruent, so their measures are equal.
   - **d. \( m∠4 = m∠3 \):**
     False. ∠4 and ∠3 form a linear pair, so their measures sum to 180°, meaning they cannot be equal.

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Would you like further clarification or details on any specific part of this?

### 5 Related Questions:
1. Can two angles form both a vertical pair and a linear pair?
2. How do adjacent angles differ from vertical angles?
3. What are supplementary and complementary angles?
4. What happens to the measures of angles when more than two lines intersect?
5. Can linear pairs ever be congruent?

### Tip:
Always remember that vertical angles are congruent, and linear pairs are supplementary!

Give two vertical angles. What can you claim about these vertical angles? Name two angles that form a linear pair. Evaluate the following angles and identify 'True' or 'False' for given angle relationships.

Explore the relationships between vertical angles and linear pairs using a detailed solution to common geometry problems. Learn how vertical angles are congruent, linear pairs are supplementary, and understand key angle properties. Perfect for students in Grades 7-10.

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