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Chapter 4 Summary

Definitions

  • The distance between two objects is the length of the shortest path between them.
  • The perpendicular bisector of a segment is the line that bisects the segment and is perpendicular to it.
  • A plane is a surface such that any two points on that surface are connected by a line, all points of which are in that surface.
  • Objects are coplanar if they all lie on the same plane.
  • Objects are noncoplanar if they are not coplanar (though a subset of them might be).
  • A transversal is a line that intersects two coplanar lines in two distinct points.
  • The interior is the region between the two coplanar lines.
  • The exterior is the region outside the two coplanar lines.
  • Alternate interior angles are on alternate sides of the transversal and on the interior of the figure.
  • Alternate exterior angles are on alternate sides of the transversal and on the exterior of the figure.
  • Corresponding angles are on the same side of a transversal and on the same side of the two lines cut by the transversal.
  • Parallel lines are coplanar lines that do not intersect.
  • Slope of a line is (y2 − y1) / (x2 − x1) or Δy/Δx

    • Postulates

    Theorems

    Midpoint formula
    Midpoint between (x1, y1) and (x2, y2) is ((x1+x2)/2, (y1+y2)/2)
    Right angle theorem
    If two angles are both supplementary and congruent, then they are right angles.
    Equidistance theorems
    If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.
    If a point is on the perpendicular bisector of a segment, then it is equidistant from the segment's endpoints.
    Slopes of parallel lines
    are equal, and lines with the same slope are parallel.
    Slopes of perpendicular lines
    are opposite reciprocals of each other. If a slope is m, its perpendicular is −1/m. Also, if the slope of one line is the opposite reciprocal of the other, then the lines are perpendicular.
    Jon Dreyer