Provingtrianglescongruentssssasasa

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Provingtrianglescongruentssssasasa

  1. 1. Proving Triangles Congruent Free powerpoints at http://www.worldofteaching.com
  2. 2. Two geometric figures with exactly the same size and shape. The Idea of a CongruenceThe Idea of a Congruence A C B DE F
  3. 3. How much do youHow much do you need to know. . .need to know. . . . . . about two triangles to prove that they are congruent?
  4. 4. In Lesson 4.2, you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. Corresponding PartsCorresponding Parts ∆ABC ≅ ∆ DEF B A C E D F 1. AB ≅ DE 2. BC ≅ EF 3. AC ≅ DF 4. ∠ A ≅ ∠ D 5. ∠ B ≅ ∠ E 6. ∠ C ≅ ∠ F
  5. 5. Do you needDo you need all six ?all six ? NO !
  6. 6. Side-Side-Side (SSS)Side-Side-Side (SSS) 1. AB ≅ DE 2. BC ≅ EF 3. AC ≅ DF ∆ABC ≅ ∆ DEF B A C E D F
  7. 7. Side-Angle-Side (SAS)Side-Angle-Side (SAS) 1. AB ≅ DE 2. ∠A ≅ ∠ D 3. AC ≅ DF ∆ABC ≅ ∆ DEF B A C E D F included angle
  8. 8. The angle between two sides Included AngleIncluded Angle ∠ G ∠ I ∠ H
  9. 9. Name the included angle: YE and ES ES and YS YS and YE Included AngleIncluded Angle SY E ∠ E ∠ S ∠ Y
  10. 10. Angle-Side-Angle-Side-AngleAngle (ASA)(ASA) 1. ∠A ≅ ∠ D 2. AB ≅ DE 3. ∠ B ≅ ∠ E ∆ABC ≅ ∆ DEF B A C E D F include d side
  11. 11. The side between two angles Included SideIncluded Side GI HI GH
  12. 12. Name the included side: ∠Y and ∠E ∠E and ∠S ∠S and ∠Y Included SideIncluded Side SY E YE ES SY
  13. 13. Angle-Angle-Side (AAS)Angle-Angle-Side (AAS) 1. ∠A ≅ ∠ D 2. ∠ B ≅ ∠ E 3. BC ≅ EF ∆ABC ≅ ∆ DEF B A C E D F Non-included side
  14. 14. Warning:Warning: No SSA PostulateNo SSA Postulate A C B D E F NOT CONGRUENT There is no such thing as an SSA postulate!
  15. 15. Warning:Warning: No AAA PostulateNo AAA Postulate A C B D E F There is no such thing as an AAA postulate! NOT CONGRUENT
  16. 16. The Congruence PostulatesThe Congruence Postulates SSS correspondence ASA correspondence SAS correspondence AAS correspondence SSA correspondence AAA correspondence
  17. 17. Name That PostulateName That Postulate SASSAS ASAASA SSSSSSSSASSA (when possible)
  18. 18. Name That PostulateName That Postulate(when possible) ASAASA SASSAS AAAAAA SSASSA
  19. 19. Name That PostulateName That Postulate(when possible) SASSAS SASSAS SASSAS Reflexive Property Vertical Angles Vertical Angles Reflexive Property SSASSA
  20. 20. Name That PostulateName That Postulate(when possible)
  21. 21. (when possible) Name That PostulateName That Postulate
  22. 22. Let’s PracticeLet’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: ∠B ≅ ∠D For AAS: ∠A ≅ ∠F AC ≅ FE
  23. 23. Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS:
  24. 24. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. ΔGIH ≅ ΔJIK by AAS G I H J K Ex 4
  25. 25. ΔABC ≅ ΔEDC by ASA B A C ED Ex 5 Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
  26. 26. ΔACB ≅ ΔECD by SAS B A C E D Ex 6 Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
  27. 27. ΔJMK ≅ ΔLKM by SAS or ASA J K LM Ex 7 Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
  28. 28. Not possible K J L T U Ex 8 Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. V
  29. 29. Hypotenuse-Leg Congruence Theorem: HL • Hypotenuse-Leg Congruence Theorem (HL) – If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and the leg of a second right triangle, then the two triangles are congruent. – If ABC and DEF are right triangles, and AC = DF, and BC = EF, then ABC = DEF ~ ~ ~ A B C D E F
  30. 30. Determine When to Use HL • Is it possible to show that the two triangles are congruent using the HL Congruence Theorem? Explain your reasoning. • The two triangles are right triangles. You know that JH = JH (Hypotenuse). You know that JG = HK (Leg). So, you can use the HL Congruence theorem to prove that JGH = HKJ. G H J K ~ ~ ~
  31. 31. Triangle Congruence Postulates and Theorems 1. SSS (Side-Side-Side) 2. SAS (Side-Angle-Side) 3. ASA (Angle-Side-Angle) 4. AAS (Angle-Angle-Side) 5. HL (Hypotenuse-Leg)

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