Parallel & Perpendicular Lines
SAT Math — Algebra & Coordinate Geometry
Master slope relationships to quickly identify parallel (∥) and perpendicular (⊥) lines on the SAT.
1. Introduction — Lines in a Plane
Two distinct lines in a plane have exactly three possible relationships:
| Relationship | Slope Condition | Symbol |
|---|---|---|
| Parallel | $m_1 = m_2$ (equal slopes) | ∥ |
| Perpendicular | $m_1 \cdot m_2 = -1$ (negative reciprocals) | ⊥ |
| Intersecting (not ⊥) | $m_1 \neq m_2$ and $m_1 \cdot m_2 \neq -1$ | — |
Same Line vs. Parallel: If $m_1 = m_2$ and $b_1 = b_2$, the equations represent the same line, not just parallel lines.
Why does this matter on the SAT?
- Geometry proofs and coordinate geometry questions
- Writing equations of lines through given points
- Solving systems of equations (parallel → no solution)
- Real-world applications (architecture, navigation, design)
- Classifying quadrilaterals (rectangles, parallelograms, trapezoids)
SAT Tip: The relationship between two lines is entirely determined by their slopes. Always find the slopes first!
Interactive Visualization: Parallel, Perpendicular & Intersecting Lines
Interactive visualization — available in the full lesson
Blue and green lines are parallel (same slope 2). Red line is perpendicular to blue (slope −½). The small square marks the 90° angle.