Lesson 5 of 10 · 7 min
Angle between two lines
NCERT §9.2.3
A second road will cross Riya's canal at a slant. If she knows the two slopes, how can she find the angle at the crossing without drawing it?
The lesson in notes
In short
Two intersecting lines make two pairs of vertically opposite angles, and adjacent angles add to 180°. If the lines have slopes m₁ and m₂, the angle θ = α₂ − α₁ between them has tan θ = (m₂ − m₁)/(1 + m₁m₂).
Taking the modulus gives the acute angle: tan θ = |(m₂ − m₁)/(1 + m₁m₂)|, provided 1 + m₁m₂ ≠ 0. The obtuse angle is 180° − θ.
If 1 + m₁m₂ = 0, the lines are perpendicular and θ = 90°.
Example: lines of slopes 1 and 0 make tan θ = 1, an angle of 45°. Lines of slopes 2 and 3 make tan θ = 1/7.
If the angle between two lines is π/4 and one slope is 1/2, then |(m − 1/2)/(1 + m/2)| = 1, which gives m = 3 or m = −1/3. There are two answers because the second line can be turned either way from the first.