Geomety

Lines and Angles

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

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.

Notation

Term

Notation

Diagram

Parallel lines

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

Parallel lines

Perpendicular lines

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

Perpendicular lines
  • 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 \(\overleftrightarrow{CG}\)
Parallel, Perpendicular, and Skew

Geometry

Lines and Angles cont.

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.
Postulates and Theorems

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.
For Perependicular Lines

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
    For Perependicular Lines theorem

    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°).
      complementary