Geometry: Proving Relationships Between Lines
Proving Relationships Between Lines
Equivalence relations are pretty important in mathematics. The usefulness of = cannot be overstated, and we've used ~= in many of our formal proofs. You might begin to think that the only relations worth talking about are equivalence relations. That would be too simple.
You know that the relations ? and ? ? are not equivalence relations. Nevertheless, these two relations are proof that you don't have to be an equivalence relation to be worthy of further study. Without a perpendicularity you could never build a kite. Without a parallelism you would never be able to set a glass down on a table without worrying about it sliding off and ruining the carpet.
A lot can happen when two parallel lines are cut by a transversal. Some very important angle relationships are just waiting to be discovered. In fact, there are some angle relationships strong enough to force two lines to be parallel. I'll start with the perpendicular, and move into the realm of the parallel.
Excerpted from The Complete Idiot's Guide to Geometry 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.