Geometry: Proving Segments and Angles Are Congruent
Proving Segments and Angles Are Congruent
After you have shown that two triangles are congruent, you can use the fact that CPOCTAC to establish that two line segments (corresponding sides) or two angles (corresponding angles) are congruent.
- Example 4: If ?R and ?V are right angles, and ?RST ~= ?VST (see Figure 12.11), write a two-column proof to show RT ~= TV.
- Solution: You need a game plan. If you could show that ?RST ~= ?VST, then you could use CPOCTAC to show that RT ~= TV. To show that ?RST ~= ?VST, you simply use the AAS Theorem.
|1.||?R and ?V are right angles, and ?RST ~= ?VST||Given|
|2.||?RST ~= ?VST||AAS Theorem|
|3.||RT; ~= TV||CPOCTAC|
- Example 5: Suppose that in Figure 12.12, ?CB bisects ?ACD and BC ? AD. Write a two-column proof to show that ?A ~= ?D.
- Solution: Because BC ? AD, you know that ?ABC ~= ?DBC. Because ?CB bisects ?ACD , you know that ?ACB ~= ?DCB. Finally, ?BC is congruent to itself, and you can use the ASA Postulate to show that ?ABC ~= ?DBC. By CPOCTAC, you can conclude that ?A ~= ?D. Let's write it up.
|1.||?CB bisects ?ACD and BC ? AD||Given|
|2.||?ABC and ?DBC are right angles||Definition of ?|
|3.||m?ABC = 90 and m?DBC = 90||Definition of right angle|
|4.||m?ABC = m?DBC||Substitution|
|5.||?ABC ~= ?DBC||Definition of|
|6.||?ACB ~= ?DCB||Definition of angle bisector|
|7.||BC ~= BC||Reflexive property of ~=|
|8.||?ABC ~= ?DBC||ASA Postulate|
|9.||?A ~= ?D||CPOCTAC|
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.