Slope and Parallel Lines

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Slope and Parallel Lines

• Sections 4.5 4.6

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Definitions

• A plane is a surface such that if any two points
  on the surface are connected by a line, all
  points of the line are also on the surface.
• A plane has only two dimensions length and
  width but no thickness.

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Definitions

• If points, lines, segments, and so forth lie in
  the same plane, we call them coplanar.
• Points, lines, segments, and so forth that do not
  lie in the same plane are called noncoplanar.

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Definitions

• A transversal is a line that intersects two
  coplanar lines in two distinct points.

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Definitions

• In the diagram, the region between lines d and e
  is the interior of the figure.
• In the diagram, the rest of the plane except the
  region between lines d and e is the exterior of
  the figure.

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Definitions

• Alternate Interior Angles are a pair of angles
  formed by two lines and a transversal. The angles
  must
• both lie in the interior of the figure,
• lie on alternate sides of the transversal,
• have different vertices.

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Definitions

• Alternate Exterior Angles are a pair of angles
  formed by two lines and a transversal. The angles
  must
• both lie in the exterior of the figure,
• lie on alternate sides of the transversal,
• have different vertices.

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Definitions

• Corresponding Angles are a pair of angles formed
  by two lines and a transversal.
• One angle must lie in the interior of the figure,
  and the other must lie in the exterior.
• The angles must lie on the same side of the
  transversal but have different vertices.

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corr. angles
alt. ext. angles
alt. int. angles
corr. angles
vert. angles
corr. angles
alt. ext. angles
alt. int. angles
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Name the pairs of...
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Parallel Lines

• Parallel () lines are two coplanar lines which
  do not intersect.
• Parallel lines have the same slope.

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Slope Review ?

• The slope of a nonvertical line (or segment or
  ray) containing points (x1, y1) and (x2, y2) is
  defined by
• Find the slope of the line containing points (2,
  -1) and (7, 4)

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Remember,

• Rising line positive slope
• Falling line negative slope
• Horizontal line zero slope
• Vertical line no slope (undefined slope)

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Slopes of Parallel Lines

• Theorem 26 If two nonvertical lines are
  parallel, then their slopes are equal.
• Theorem 27 If the slopes of two nonvertical
  lines are equal, then the lines are parallel.

Note You do NOT need to prove these theorems!!
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Slopes of Perpendicular Lines
Note You do NOT need to prove these theorems!!

• Theorem 28 If two nonvertical lines are
  perpendicular, then each lines slope is the
  opposite reciprocal of the others.
• Theorem 29 If a lines slope is the opposite
  reciprocal of another lines slope, then the two
  lines are perpendicular.

Flip the top and bottom of fraction and change to
opposite sign!