Geometry/Parallel and Perpendicular Lines and Planes
Parallel Lines in a Plane
Two coplanar lines are said to be parallel if they never intersect. For any given point on the first line, its distance to the second line is equal to the distance between any other point on the first line and the second line. The common notation for parallel lines is "||" (a double pipe); it is not unusual to see "//" as well. If line m is parallel to line n, we write "m || n". Lines in a plane either coincide, intersect in a point, or are parallel. Controversies surrounding the Parallel Postulate lead to the development of non-Euclidean geometries.
Parallel Lines and Special Pairs of Angles
When two (or more) parallel lines are cut by a transversal, the following angle relationships hold:
- corresponding angles are congruent
- alternate exterior angles are congruent
- same-side interior angles are supplementary
Theorems Involving Parallel Lines
- If a line in a plane is perpendicular to one of two parallel lines, it is perpendicular to the other line as well.
- If a line in a plane is parallel to one of two parallel lines, it is parallel to both parallel lines.
- If three or more parallel lines are intersected by two or more transversals, then they divide the transversals proportionally.