Parallel and perpendicular lines
Parallel lines lie in the same plane and never intersect; perpendicular lines intersect at a 90° angle. In coordinate geometry, parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other.
Parallel lines are lines in the same plane that never meet, no matter how far they extend — they keep a constant distance apart. Perpendicular lines are lines that intersect at a right angle (90°). These two relationships are the backbone of coordinate geometry questions on standardized tests.
On the coordinate plane, the relationships come down to slope. Parallel lines have the same slope but different y-intercepts: y = 2x + 1 and y = 2x − 7 never intersect. Perpendicular lines have slopes that are negative reciprocals — their slopes multiply to −1. A line with slope 2 is perpendicular to any line with slope −1/2; a horizontal line (slope 0) is perpendicular to a vertical line (undefined slope). A typical problem gives you a line such as y = 3x + 4 and a point, and asks for the equation of the parallel or perpendicular line through that point.
When a third line, called a transversal, cuts across two parallel lines, it creates predictable angle pairs: corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary (sum to 180°). These angle facts let you chase unknown angles through a figure from a single given measurement.
Parallel and perpendicular lines appear throughout the GRE quantitative section, the CLT's algebra questions on linear equations, and AMC competition geometry. Know the slope rules cold and be ready to combine them with transversal angle relationships.
Key takeaways
- Parallel lines never intersect and have equal slopes; perpendicular lines meet at 90° and have negative reciprocal slopes.
- Two lines are perpendicular exactly when the product of their slopes is −1 (or one is horizontal and the other vertical).
- A transversal across parallel lines makes corresponding and alternate interior angles equal, and same-side interior angles supplementary.
- A classic test question asks for the line through a given point that is parallel or perpendicular to a given line.
- The GRE, CLT, and AMC all test slope relationships and parallel-line angle facts.