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.

Figure 12.11∠R and ∠V are right angles, and ∠RST ~= ∠VST.

  • 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 ~= ΔVSTAAS 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.

Figure 12.12→CB bisects ∠ACD and ¯BC ⊥ ¯AD.

  • 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 anglesDefinition 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.

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