Some lines intersect and some lines don’t. Imagine a set of railroad tracks. The two tracks in the set never intersect, so they’re parallel. At a crossroad, however, another pair of tracks intersects these tracks, forming 90° angles. The two sets of tracks are perpendicular. There’s a power line that crosses above the tracks, but the power line and the tracks never intersect because the power line is above the tracks - they’re on different planes. Lines work the same way as railroad tracks, except that lines can intersect in many different ways to produce angles with various properties.
Parallel Lines: Lines that lie in the same plane and never intersect.
Parallel Planes: Planes that lie in the same plane and never intersect.
Perpendicular Lines: Lines that intersect to form four right angles.
Perpendicular Planes: Planes that intersect to form four right angles.
Skew Lines: Lines that lie in different planes and never intersect.
Perpendicular Bisector: A line, ray, or segment that passes through the midpoint of a segment at a right angle.
Parallel lines
\(\overleftrightarrow{AB} || \overleftrightarrow{MN} \text{ or / } || m\)
Perpendicular lines
\(\overleftrightarrow{AS} \perp \overleftrightarrow{TC}\)
Think of each segment in the figure as part of a line. Examples:
Parallel Lines
Parallel Planes
Perpendicular Lines
Perpendicular Planes
Skew Lines
Parallel Line Postulate: Given a line and a point not on the line, there is exactly one line parallel to this line that goes through the point.
Transitive Property of Parallel Lines: If two lines are parallel to the same line, then they are parallel to each other.
Perpendicular Line Postulate: Given a line and a point not on the line, there is exactly one line perpendicular to this line that goes through the point.
A similar postulate is the Perpendicular Bisector Postulate.
Perpendicular Bisector Postulate: For every line segment, there is one perpendicular bisector that passes through the midpoint.
Theorems: