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Parallel Lines and Planes Help

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Introduction to Geometry Rules for LInes and Angles of a Plane

In the confines of a single geometric plane, lines and angles behave according to various rules. The following are some of the best-known examples.

Parallel Planes

Two distinct planes are parallel if and only if they do not intersect. Two distinct half planes are parallel if and only if the planes in which they lie do not intersect. Two distinct plane regions are parallel if and only if the planes in which they lie do not intersect.

Distance Between Parallel Planes

Suppose planes X and Y are parallel. Let R be some arbitrary point on plane X . Then the distance between planes X and Y is equal to the distance between point R and plane Y , which has already been defined.

Vertical Angles For Intersecting Planes

Suppose that Y and Z are two planes that intersect in a line L . Let points P, Q, R, S , and T be as shown in Fig. 7-14, such that:

  • Point T lies on lines L, PS , and RQ
  • Points Q and R lie in plane Y
  • Points P and S lie in plane Z
  • Lines PS and RQ are both perpendicular to line L

 

An Expanded Set of Rules More Facts Vertical Angles For Intersecting Planes

Fig. 7-14 . Vertical angles between intersecting planes.

Then the pair ∠PTQ and ∠STR are vertical angles; the pair ∠RTP and ∠QTS are also vertical angles. Thus, ∠PTQ has the same measure as ∠STR , and ∠RTP has the same measure as ∠QTS .

Alternate Interior Angles For Intersecting Planes

Suppose that X is a plane that passes through two parallel planes Y and Z , intersecting Y and Z in lines L and M . Let points P, Q, R, S, T, U, V , and W be as shown in Figs. 7-15A and 7-15B, such that:

 

 

An Expanded Set of Rules More Facts Alternate Interior Angles For Intersecting Planes

An Expanded Set of Rules More Facts Alternate Interior Angles For Intersecting Planes

Fig. 7-15 . (A) Alternate interior angles between intersecting planes. (B) Another example of alternate interior angles between intersecting planes.

  • Point V lies on lines L, PQ , and RS
  • Point W lies on lines M, PQ , and TU
  • Points P and Q lie in plane X
  • Points R and S lie in plane Y
  • Points T and U lie in plane Z
  • Lines PQ and RS are perpendicular to line L
  • Lines PQ and TU are perpendicular to line M

Then the pair ∠PVR and ∠QWU are alternate interior angles (Fig. 7-15A); the pair ∠TWQ and ∠SVP are also alternate interior angles (Fig. 7-15B). Alternate interior angles always have equal measures. Thus, ∠PVR has the same measure as ∠QWU , and ∠TWQ has the same measure as ∠SVP .

Alternate Exterior Angles For Intersecting Planes

Suppose that X is a plane that passes through two parallel planes Y and Z , intersecting Y and Z in lines L and M . Let points P, Q, R, S, T, U, V , and W be as shown in Fig. 7-16, such that:

  • Point V lies on lines L, PQ , and RS
  • Point W lies on lines M, PQ , and TU
  • Points P and Q lie in plane X
  • Points R and S lie in plane Y
  • Points T and U lie in plane Z
  • Lines PQ and RS are perpendicular to line L
  • Lines PQ and TU are perpendicular to line M

Then the pair ∠PWT and ∠QVS are alternate exterior angles; the pair ∠UWP and ∠RVQ are also alternate exterior angles. Alternate exterior angles always have equal measures. Thus, ∠PWT has the same measure as ∠QVS , and ∠UWP has the same measure as ∠RVQ .

 

An Expanded Set of Rules More Facts Alternate Exterior Angles For Intersecting Planes

Fig. 7-16 . Alternate exterior angles between intersecting planes.

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