Big Picture

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.

Key Terms

Notation

Term

Notation

Diagram

Parallel lines

\(\overleftrightarrow{AB} || \overleftrightarrow{MN} \text{ or / } || m\)

Perpendicular lines

\(\overleftrightarrow{AS} \perp \overleftrightarrow{TC}\)

• To mark parallel lines in a diagram, draw arrows (>) on each parallel line. Lines that are parallel will have the same number of arrows.
• To mark perpendicular lines in a diagram, draw a half-square marking the angle.
• Only one of the right angles needs to be marked with the half-square.

Parallel, Perpendicular, and Skew

Think of each segment in the figure as part of a line. Examples:

Parallel Lines

• \(\overleftrightarrow{AB} || \overleftrightarrow{EF}, \overleftrightarrow{AC} || \overleftrightarrow{BD}\)

Parallel Planes

• Plane \(\overleftrightarrow{AEB} ||\) Plane \(\overleftrightarrow{CGH}\), Plane \(\overleftrightarrow{AEG} ||\) Plane \(\overleftrightarrow{BFH}\)

Perpendicular Lines

• \(\overleftrightarrow{AB} \perp \overleftrightarrow{AC}, \overleftrightarrow{BD} \perp \overleftrightarrow{DH}\)

Perpendicular Planes

• plane \(\overleftrightarrow{AEB} \perp\) plane \(\overleftrightarrow{CAD}\), plane \(\overleftrightarrow{CGD} \perp\)  plane \(\overleftrightarrow{AEB}\)

Skew Lines

• \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{EG}\), \(\overleftrightarrow{BD}\) and \(\overleftrightarrow{CG}\)

Postulates and Theorems

For Parallel 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.

• There are infinitely many lines that pass through point P, but only one of those lines will be parallel to line l.

Transitive Property of Parallel Lines: If two lines are parallel to the same line, then they are parallel to each other.

• If lines l || m and m || n, then l || n.

For Perependicular Lines

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.

• There are infinitely many lines that pass through point P, but only one of those lines will be perpendicular to line l.

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.

• \(\overleftrightarrow{CD}\) is the perpendicular bisector of AB and divides the segments into \(\overline{AD}\) and \(\overline{DB}\).  AD = DB

Theorems:

• If two lines are perpendicular, they intersect to form four right angles.
• If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
        a. If \(m \perp l\), then  ∠∠∠1 and  ∠2 are complementary (add up to 90°).