prove that two lines m and n be parallel to the same given line are parallel to each other.

Answer

To prove that two lines \( m \) and \( n \), which are both parallel to the same line \( l \), are parallel to each other, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have two lines, \( m \) and \( n \).
- Both lines are parallel to a third line \( l \).
- Therefore, we can write: \( m \parallel l \) and \( n \parallel l \).

**Hint**: Remember that parallel lines do not intersect and maintain a constant distance from each other.

2. **Introduce a Transversal**:
- To analyze the angles formed by these lines, we introduce a transversal line \( t \) that intersects lines \( m \), \( n \), and \( l \).

**Hint**: A transversal line creates angles that can help us establish relationships between the lines.

3. **Identify Corresponding Angles**:
- When the transversal \( t \) intersects line \( m \), it creates angles. Let's denote the angles formed at the intersection of \( m \) and \( t \) as angle 1 and angle 2.
- Since \( m \parallel l \), angle 1 and angle 2 are corresponding angles and thus are equal:
\[ \text{Angle 1} = \text{Angle 2} \]

**Hint**: Corresponding angles are formed when a transversal crosses parallel lines and are located in the same relative position.

4. **Identify More Corresponding Angles**:
- Now, when the transversal \( t \) intersects line \( n \), it creates another angle, which we can call angle 3.
- Since \( n \parallel l \), angle 2 (formed at line \( m \)) and angle 3 (formed at line \( n \)) are also corresponding angles and are equal:
\[ \text{Angle 2} = \text{Angle 3} \]

**Hint**: Again, look for the same relative positions of angles when a transversal crosses parallel lines.

5. **Establish the Equality of Angles**:
- From the previous steps, we have:
\[ \text{Angle 1} = \text{Angle 2} \quad \text{and} \quad \text{Angle 2} = \text{Angle 3} \]
- By the transitive property of equality, we can conclude:
\[ \text{Angle 1} = \text{Angle 3} \]

**Hint**: The transitive property states that if \( a = b \) and \( b = c \), then \( a = c \).

6. **Conclude that Lines are Parallel**:
- Since angle 1 and angle 3 are corresponding angles and they are equal, we can conclude that lines \( m \) and \( n \) are parallel to each other:
\[ m \parallel n \]

**Hint**: The converse of the corresponding angles postulate states that if two lines are cut by a transversal and the corresponding angles are equal, then the lines are parallel.

### Final Conclusion:
Thus, we have proven that if two lines \( m \) and \( n \) are both parallel to the same line \( l \), then lines \( m \) and \( n \) are parallel to each other.