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 ΔRST ~= ΔVST, then you could use CPOCTAC to show that ¯RT ~= ¯TV. To show that ΔRST ~= ΔVST, you simply use the AAS Theorem.
|1.||∠R and ∠V are right angles, and ∠RST ~= ∠VST||Given|
|2.||ΔRST ~= ΔVST||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 ΔABC ~= ΔDBC. 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 = 90º and m∠DBC = 90º||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.||ΔABC ~= ΔDBC||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.