Slopes of Parallel Lines
Parallel lines in a coordinate plane have the same slope. This simple rule connects the geometry of parallelism to coordinate algebra.
Introduction
In Chapter 3.1 you proved lines parallel using angle relationships. Now you can prove lines parallel on the coordinate plane using a much faster test: compare their slopes.
Past Knowledge
Slope formula (3.2.1). Parallel lines definition (3.1.1).
Today's Goal
Determine whether two lines are parallel by comparing slopes.
Future Success
Writing equations of parallel lines (3.2.4–3.2.5) and triangle proofs with coordinates.
Key Concepts
Parallel Lines Slope Theorem
Two non-vertical lines are parallel if and only if they have the same slope: .
- All vertical lines are parallel to each other (both have undefined slope).
- All horizontal lines are parallel to each other ().
- Same slope ≠ same line. Lines must also have different y-intercepts to be distinct parallel lines.
Interactive Diagram — Desmos Geometry
Two lines with the same slope but different y-intercepts — they are parallel.
Worked Examples
Are They Parallel?
Line 1 passes through and . Line 2 passes through and . Are they parallel?
Yes, parallel. .
From Equations
Are and parallel?
Rewrite the second: →
Slopes: ,
Yes, parallel. Same slope, different y-intercepts (1 ≠ −4).
Finding k for Parallel Lines
Line through and is parallel to a line with slope . Find .
Common Pitfalls
Same Slope = Same Line?
Not necessarily! If two lines also have the same y-intercept, they are the same line, not parallel. Parallel lines have equal slopes and different intercepts.
Real-Life Applications
Multi-Lane Highways
Each lane of a highway is a line with the same slope (grade). Since parallel lines have equal slopes, every lane rises at the same rate — keeping cars level on banked curves and hills.
Practice Quiz
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