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

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

- 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.

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 \(\overleftrightarrow{CG}\)

*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.

*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°).