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3.Angle Between Lines
Class 11 • Mathematics • NCERT
Angle Between Two Lines
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This lesson explains how to find the angle between two straight lines using their slopes as per NCERT Class 11. Students learn the standard formula, special cases, and solve simple exam-oriented problems.
Lesson Objectives
- Understand the angle between two straight lines.
- Use slopes to calculate the angle.
- Identify parallel and perpendicular lines.
- Solve NCERT-style problems.
1. Meaning of Angle Between Two Lines
The angle between two straight lines is the angle through which one line must be rotated to coincide with the other.
The angle is usually taken as the acute angle between the lines.
2. Formula for Angle Between Two Lines
If the slopes of two lines are m₁ and m₂, then the angle θ between them is given by:
tan θ = |(m₂ − m₁) / (1 + m₁m₂)|
3. Parallel Lines
Two lines are parallel if their slopes are equal.
Condition for parallel lines:
m₁ = m₂ → θ = 0°
m₁ = m₂ → θ = 0°
4. Perpendicular Lines
Two lines are perpendicular if the product of their slopes is −1.
Condition for perpendicular lines:
m₁m₂ = −1 → θ = 90°
m₁m₂ = −1 → θ = 90°
5. Simple NCERT Examples
• Find angle between lines with slopes 1 and √3
tan θ = |(√3 − 1)/(1 + √3)|
tan θ = |(√3 − 1)/(1 + √3)|
• If m₁ = 2 and m₂ = −1/2, lines are perpendicular
Practice Questions (NCERT)
- Define the angle between two lines.
- Write the formula for angle between two lines.
- Find the angle between lines with slopes 1 and −1.
- What is the condition for two lines to be parallel?
- What is the condition for two lines to be perpendicular?
- What is the angle between parallel lines?
- What is the angle between perpendicular lines?
- If m₁ = m₂, what is the value of θ?
- If m₁m₂ = −1, what is θ?
- Is this topic part of NCERT Class 11 syllabus?
✅ Show Answer Key
- The angle through which one line must rotate to coincide with another.
- tan θ = |(m₂ − m₁)/(1 + m₁m₂)|
- 90°
- m₁ = m₂
- m₁m₂ = −1
- 0°
- 90°
- 0°
- 90°
- Yes
© Aviate Learning – Straight Lines (NCERT Class 11)