Two angles are congruent if they have the same measure. You already know that when two lines intersect the vertical angles formed are congruent. You have also seen that if A and B are each complementary to C, then A ~= B. There are other angle relationships to explore. When you expose these angle relationships, you will establish their truth using a formal proof. For example, you were introduced to the idea of an angle bisector. Well, it turns out that the bisector of an angle divides the angle into two angles, each of which has measure equal to one-half the measure of the original angle. This statement looks a lot like Theorem 9.1 applied to angles rather than segments. You can use a game plan similar to the one you used to prove Theorem 9.1 to prove this theorem.
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. To order this book direct from the publisher, visit the Penguin USA website or call 1-800-253-6476. You can also purchase this book at Amazon.com and Barnes & Noble.
When triangles are congruent, one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and angles will be congruent.
The good news is that when proving triangles congruent, it is not necessary to prove all six facts to show congruency. There are certain ordered combinations of these facts that are sufficient to prove triangles congruent. These combinations guarantee that, given these facts, it will be possible to draw triangles which will take on only one shape (be unique), thus insuring congruency.
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