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Geometry

Geometry. 5.6 Proving Triangles Congruent using ASA and AAS. EQ: What other ways can you prove triangles congruent?. Goals. Use ASA and AAS postulates to prove two triangles are congruent. Solve problems using congruence postulates. AC is included between A and C.

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Geometry

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  1. Geometry 5.6 Proving Triangles Congruent using ASA and AAS EQ: What other ways can you prove triangles congruent?

  2. Goals • Use ASA and AAS postulates to prove two triangles are congruent. • Solve problems using congruence postulates. Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  3. AC is included between A and C. AB is included between A and B. CB is included between C and B. Included Sides C B A Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  4. C Z A B X Y Theorem 5.10: Angle-Side-Angle If two angles and the included side of one triangle are congruent to corresponding parts of another triangle, then the triangles are congruent. (ASA) ABC  XYZ Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  5. DB  DB (Ref.) ExampleProve ADB  CBD C D Proof: ADB  CBD (Given) ABD  CDB (Given) ADB  CBD (ASA) A S A B A Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  6. E B F C D A Theorem 5.11 Angle-Angle-Side • If two angles and a nonincluded side of one triangle are congruent to corresponding parts of another triangle, then the triangles are congruent. ABC  DEF Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  7. A S B D C ExampleIs ABC  ADC? Yes,by AAS A A Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  8. Are these triangles congruent? Yes. Reason: AAS OR AAS. Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  9. Are these triangles congruent? You could even use ASA. Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  10. C C B B D D D D A A A A Overlapping Triangles Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  11. Problem: Is ABD  DCA? C B 85° 85° 15° 15° D A YES by AAS Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  12. B C 85° 85° 15° 15° A D Problem: Is ABD  DCA? C B 85° 85° 15° 15° D A YES by AAS Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  13. Methods to show triangles congruent: • SSS • SAS • ASA • AAS When trying to determine if triangles are congruent, carefully study the given information, mark the triangles if necessary, then determine what pattern you have. Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  14. BC  ST T Problem A S S A B S A C R A A What other congruence must be given to show that ABC  RST using the AAS postulate? Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  15. BA  SR T Alternate Solution Problem A S B S A A C R S A A What other congruence must be given to show that ABC  RST using the AAS postulate? Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  16. AC  RT T Problem A B S A S C A R A S A What other congruence must be given to show that ABC  RST using the ASA postulate? Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  17. State the third congruence that must be given to prove the given triangles congruent using the indicated postulate or theorem. Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  18. B F D C A E 1. ASA Theorem. C  F Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  19. D A E B C F 2. AAS Theorem. C  F Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  20. B D C F A E CB  FE 3. SSS Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  21. 4. SAS Theorem. F E A D C B A  D Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  22. Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem you would use. Explain reasoning Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  23. W X Z 5. Yes AAS Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  24. 6. C A Yes SAS D R Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  25. T S O L F 7. The only other thing we know for sure: SOT  LOF (Vertical Angles) But this spells… SSA NOT CONGRUENT Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  26. 1. LA || SN L S 3. LR  NR R A N Given: LA || SN, LR  NR Prove: LAR  NSR 1. Given 2. L  N 2. Alt. int s 3. Given 4. LRA  NRS 4. Vert s 5. LAR  NSR 5. ASA Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  27. List four ways to show triangles are congruent. • SSS • ASA • SAS • AAS List one way that won’t work. SSA Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  28. Important! • To show two triangles congruent, at least one side must be known. • Three angles (AAA) does not show congruence (but it is still useful). Geometry 5.6 Proving Triangles Congruent: ASA, AAS

  29. Summarize Geometry 5.6 Proving Triangles Congruent: ASA, AAS

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