Equation of a Line
306. Slopes of Parallel Lines · equal slopes, forever parallel
The slopes of two parallel lines are always equal.
Parallel lines have equal slopes. Since the slope fixes a line's direction, two lines point the same way exactly when m₁ = m₂ (and so never meet).
What this lesson covers
Try to break it
Drag S around its circle. Both lines rotate together by the same angle, keeping θ = α. Since slope m = tan(angle), and both angles match, the two slopes match too — that's why parallel lines always have equal slopes.
How you build it
Draw a line and a parallel to it, then compare their slopes.
- Using the Line tool, draw a slanted line AB across the grid. Count its rise over run — that is its slope m₁.
- With the Parallel tool, click a point away from AB, then click AB. The new line CD has the same rise and run as AB, so m₂ = m₁ — equal slopes, forever parallel.
The proof, step by step
Prove that two parallel lines have equal slopes.
- Given: Two straight lines AB and CD are parallel to each other.
- Draw a transversal intersecting AB and CD. Let the corresponding angles be θ and α.
- Since the lines are parallel, the corresponding angles are equal, i.e., θ = α.
- Therefore, tan θ = tan α. Hence, slope of AB = slope of CD.
Worked example
Line L1 passes through points (1, 2) and (3, 6). Line L2 is parallel to L1. What is the slope of L2?
Slope of L1 = (6-2)/(3-1) = 4/2 = 2. Since L2 is parallel to L1, its slope is also 2.
- 1
- 2 — correct
- 3
- 4