Relationships Between Lines

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Relationships Between Lines
Parallel Lines two lines that are coplanar and
do not intersect Skew Lines two lines that
are NOT coplanar and do not intersect Perpendicul
ar Lines two lines that intersect to form a
right angle Parallel Planes two planes that do
not intersect
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Parallel and Perpendicular Postulates
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• Postulate 13 Parallel Postulate
• If there is a line and a point not on the line,
  then there is exactly one line through the point
  parallel to the given line.
• Postulate 14 Perpendicular Postulate
• If there is a line and a point not on the line,
  then there is exactly one line through the point
  perpendicular to the given point.

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Transversals

• A transversal is a line that intersects two or
  more coplanar lines at different points.

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Angles Formed By Transversals
Corresponding Angles two angles are
corresponding if they occupy corresponding
positions as a result of the intersection of the
lines. (1 5) Alternate Exterior Angles two
angles are alternate exterior if they lie outside
the two lines on opposite sides of the
transversal (1 7) Alternate Interior Angles
Two angles that lie between the two lines on
opposite sides of the transversal (4
6) Consecutive Interior Angles two angles that
lie between the two lines on the same side of the
transversal (4 5)
Who can name another pair for each of the angle
types above?
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Postulates and Theorems of Parallel Lines

• Postulate 15 Corresponding Angles Postulate
• If two parallel lines are cut by a transversal,
  then the pairs of corresponding angles are
  congruent.
• Theorem 3.4 Alternate Interior Angles
• If two parallel lines are cut by a transversal,
  then the pairs of alternate interior angles are
  congruent.
• Theorem 3.5 Consecutive Interior Angles
• If two parallel lines are cut by a transversal,
  then the pairs of consecutive interior angles are
  supplementary.
• Theorem 3.5 Alternate Exterior Angles
• If two parallel lines are cut by a transversal,
  then the pairs of alternate exterior angles are
  congruent.
• Theorem 3.7 Perpendicular Transversal
• If a transversal is perpendicular to one of two
  parallel lines, then it is perpendicular to the
  other.

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Proving the Alternate Interior Angles Theorem
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Proving Lines are Parallel
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Postulate 16 Corresponding Angle Converse If
two lines are cut by a transversal so that the
corresponding angles are congruent, then the
lines are parallel. Theorem 3.8 Alternate
Interior Angles Converse If two lines are cut by
a transversal so that the alternate interior
angles are congruent, then the lines are
parallel. Theorem 3.9 Consecutive Interior
Angles Converse If two lines are cut by a
transversal so that the consecutive interior
angles are supplementary, then the lines are
parallel. Theorem 3.10 Alternate Exterior Angles
Converse - If two lines are cut by a transversal
so that alternate exterior angles are congruent,
then the lines are parallel.
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Proving the Alternate Interior Angles Converse
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