Geometry For Dummies, 3rd Edition
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When lines and planes are perpendicular and parallel, they have some interesting properties. You can use some of these properties in 3-D proofs that involve 2-D concepts, such as proving that you have a particular quadrilateral or proving that two triangles are similar.
  • Three parallel planes: If two planes are parallel to the same plane, then they're parallel to each other.
  • Two parallel lines and a plane:
    • If two lines are perpendicular to the same plane, then they're parallel to each other.

      geometry-parallel-lines

    • If a plane is perpendicular to one of two parallel lines, then it's perpendicular to the other.

      geometry-perpendicular

  • Two parallel planes and a line:
    • If two planes are perpendicular to the same line, then they're parallel to each other.

      geometry-prallel-planes

    • If a line is perpendicular to one of two parallel planes, then it's perpendicular to the other.

      geometry-perpendicular-planes

And here's a theorem you need for the example problem that follows.

A plane that intersects two parallel planes: If a plane intersects two parallel planes, then the lines of intersection are parallel. Note: Before you use this theorem in a proof, you usually have to show that the plane that cuts the parallel planes is, in fact, a plane.

geometry-plane-intersection

geometry-eq

Here's the proof diagram.

geometry-perpendicular-diagram

Here's the final proof:

geometry-perpendicular-proof

About This Article

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About the book author:

Mark Ryan is the founder and owner of The Math Center in the Chicago area, where he provides tutoring in all math subjects as well as test preparation. Mark is the author of Calculus For Dummies, Calculus Workbook For Dummies, and Geometry Workbook For Dummies.

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