Angle Between Two Lines — Formula and Examples

If two non-vertical lines have slopes m₁ and m₂, and θ is the acute angle between them, the formula is:

tan θ = |(m₁ − m₂) / (1 + m₁m₂)|

Where:

• θ is the acute angle (0° ≤ θ < 90°)
• m₁ is the slope of the first line
• m₂ is the slope of the second line
• The absolute value bars | | ensure the result is positive, giving the acute angle.

The angle formula gives us two extremely important conditions:

1. Parallel Lines (θ = 0°) If the lines are parallel, the angle between them is 0°. Since tan 0° = 0: (m₁ − m₂) / (1 + m₁m₂) = 0 m₁ = m₂ Rule: Parallel lines have equal slopes.

2. Perpendicular Lines (θ = 90°) If the lines are perpendicular, the angle is 90°. Since tan 90° is undefined (denominator must be 0): 1 + m₁m₂ = 0 m₁m₂ = −1 Rule: The product of slopes of perpendicular lines is -1 (they are negative reciprocals).

If the lines are given as vectors in 3D space: Line 1 direction vector = a Line 2 direction vector = b

The angle θ between them is found using the dot product: cos θ = |a · b| / (|a| |b|)

Question: Find the acute angle between lines with slopes 2 and 1/3.

Solution: m₁ = 2, m₂ = 1/3

tan θ = |(m₁ − m₂) / (1 + m₁m₂)| tan θ = |(2 − 1/3) / (1 + 2(1/3))| tan θ = |(5/3) / (1 + 2/3)| tan θ = |(5/3) / (5/3)| tan θ = 1

Since tan θ = 1, the acute angle θ = 45° (or π/4 radians).