If two straight line are perpendicular to the same line, prove that they are parallel to each other.

To prove that two straight lines that are both perpendicular to the same line are parallel to each other, we can use the concept of angles formed by perpendicular lines.

### Given:
- Let line \( l_1 \) and line \( l_2 \) be two straight lines.
- Let line \( m \) be a third line that is perpendicular to both \( l_1 \) and \( l_2 \).

### To Prove:
- \( l_1 \) is parallel to \( l_2 \).

### Proof:
1. **Perpendicularity Definition**:
   - Since \( l_1 \) is perpendicular to \( m \), the angle between \( l_1 \) and \( m \) is \( 90^\circ \). So, \( \angle (l_1, m) = 90^\circ \).
   - Similarly, since \( l_2 \) is perpendicular to \( m \), the angle between \( l_2 \) and \( m \) is \( 90^\circ \). So, \( \angle (l_2, m) = 90^\circ \).

2. **Equal Corresponding Angles**:
   - The angles \( \angle (l_1, m) \) and \( \angle (l_2, m) \) are equal because both are \( 90^\circ \).

3. **Parallel Lines Condition**:
   - According to the parallel lines postulate, if a transversal (in this case, line \( m \)) forms equal corresponding angles with two lines, then those two lines must be parallel.
   - Here, \( l_1 \) and \( l_2 \) have equal corresponding angles with the transversal \( m \).

4. **Conclusion**:
   - Therefore, \( l_1 \) and \( l_2 \) must be parallel.

Hence, we have proved that two straight lines that are perpendicular to the same line are parallel to each other.

Would you like a more detailed explanation or have any questions?

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Here are 5 related questions:
1. What is the converse of the statement that parallel lines have equal corresponding angles?
2. How can the concept of perpendicularity be used to solve geometric problems involving triangles?
3. If two lines are both parallel to a third line, what can be said about their relationship to each other?
4. How does the concept of perpendicular bisectors apply in the construction of geometric shapes?
5. What is the relationship between perpendicular lines and the angles they form with other intersecting lines?

**Tip:** In geometry, understanding the relationship between angles formed by intersecting lines is crucial for solving problems involving parallel and perpendicular lines.

Learn how to prove that two straight lines perpendicular to the same line are parallel to each other. This geometric problem uses the concept of perpendicularity and parallel lines postulate. Suitable for high school students studying geometry.

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