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Use the given figure to find the measures of the numbered​ angles, given that lines m and n are parallel.

To solve this problem, we will analyze the angles in the figure based on the given information that lines \( m \) and \( n \) are parallel. We'll use the properties of parallel lines and transversals to determine the measures of the numbered angles.

### Observations:
- When a transversal crosses parallel lines, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.

### Given Figure Information:
- The lines \( m \) and \( n \) are parallel.
- There are numbered angles in the diagram (1, 2, 3, 4, 5, 6).

### Step-by-Step Solution:

1. **Identify Corresponding Angles:**
   - Angle 1 and Angle 5 are corresponding angles because they are on the same side of the transversal and the lines \( m \) and \( n \) are parallel.
   - Therefore, \( \text{Angle 1} = \text{Angle 5} \).

2. **Identify Alternate Interior Angles:**
   - Angle 1 and Angle 4 are alternate interior angles.
   - Therefore, \( \text{Angle 1} = \text{Angle 4} \).
   - Angle 3 and Angle 6 are also alternate interior angles.
   - Therefore, \( \text{Angle 3} = \text{Angle 6} \).

3. **Identify Vertical Angles:**
   - Angle 2 and Angle 4 are vertical angles.
   - Therefore, \( \text{Angle 2} = \text{Angle 4} \).
   - Similarly, Angle 3 and Angle 5 are vertical angles.
   - Therefore, \( \text{Angle 3} = \text{Angle 5} \).

4. **Determine Relationships:**
   - Since Angle 1 = Angle 4, and Angle 2 = Angle 4, we can conclude that Angle 1 = Angle 2.
   - Since Angle 3 = Angle 6, and Angle 3 = Angle 5, we can conclude that Angle 6 = Angle 5.

5. **Supplementary Angles:**
   - Angle 1 and Angle 3 are on the same side of the transversal and therefore are supplementary. 
   - \( \text{Angle 1} + \text{Angle 3} = 180^\circ \).

### Conclusion:
- If any angles are given numerically, we can use these relationships to determine the specific measures of the numbered angles.

Please let me know if you would like more detailed calculations or if you have any specific angles provided numerically.

### Questions for Further Exploration:
1. How do you find the measure of an angle if only one angle is given numerically?
2. What are the properties of supplementary angles in the context of parallel lines and transversals?
3. How do vertical angles relate to each other in a parallel line system?
4. Can the relationship between corresponding angles be used to solve real-world geometry problems?
5. How does understanding alternate interior angles help in proving two lines are parallel?

### Tip:
Always remember that corresponding, alternate interior, and vertical angles are key when working with parallel lines and transversals.

Discover how to determine angle measures in a diagram where lines m and n are parallel. Learn about corresponding angles, alternate interior angles, and vertical angles, crucial for understanding geometric relationships in parallel lines. Suitable for students in Grades 8-10.

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