2. Lessons 4-3 and 4-4 Visit www.worldofteaching.comwww.worldofteaching.com For 100’s of free powerpoints. This Powerpoint has been modified by Lisa Palen from a Powerpoint found at www.worldofteaching.com and a Powerpoint found at IGO Geometry Onlinewww.worldofteaching.comIGO Geometry Online Proving Triangles Congruent

3. Two geometric figures with exactly the same size and shape. The Idea of Congruence A C B DE F

4. How much do you need to know... need to know...... about two triangles to prove that they are congruent?

5. 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 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

6. Do you need all six ? NO ! SSS SAS ASA AAS HL

7. Side-Side-Side (SSS) 1. AB  DE 2. BC  EF 3. AC  DF  ABC   DEF B A C E D F Side The triangles are congruent by SSS. If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

8. The angle between two sides Included Angle HGIHGI GG GIHGIH II GHIGHI HH This combo is called side-angle-side, or just SAS.

9. Name the included angle: YE and ES ES and YS YS and YE Included Angle SY E  YES or  E  YSE or  S  EYS or  Y The other two angles are the NON-INCLUDED angles.

10. Side-Angle-Side (SAS) 1. AB  DE 2.  A   D 3. AC  DF  ABC   DEF B A C E D F included angle Side Angle Side The triangles are congruent by SAS. If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.

11. The side between two angles Included Side GI HI GH This combo is called angle-side-angle, or just ASA.

12. Name the included side:  Y and  E  E and  S  S and  Y Included Side SY E YE ES SY The other two sides are the NON-INCLUDED sides.

13. Angle-Side- Angle (ASA) 1.  A   D 2. AB  DE 3.  B   E  ABC   DEF B A C E D F included side Angle Side Angle The triangles are congruent by ASA. If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.

14. E D F Angle-Angle-Side (AAS) 1.  A   D 2.  B   E 3. BC  EF  ABC   DEF Non-included side B A C Side Angle The triangles are congruent by AAS. If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent.

15. Warning: No SSA Postulate There is no such thing as an SSA postulate! The triangles are NOTcongruent! Side Angle

16. Warning: No SSA Postulate NOT CONGRUENT! There is no such thing as an SSA postulate!

17. BUT: SSA DOES work in one situation! If we know that the two triangles are right triangles! Side Angle

18. We call this These triangles ARE CONGRUENT by HL! HL, for “Hypotenuse – Leg” Hypotenuse Leg Hypotenuse RIGHT Triangles! Remember! The triangles must be RIGHT!

19. Hypotenuse-Leg (HL) 1.AB  HL 2.CB  GL 3.  C and  G are rt.  ‘s  ABC   DEF The triangles are congruent by HL. Right Triangle Leg Hypotenuse If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

20. Warning: No AAA Postulate A C B D E F There is no such thing as an AAA postulate! NOT CONGRUENT! Same Shapes! Different Sizes!

21. Congruence Postulates and Theorems SSS SAS ASA AAS AAA? SSA? HL

22. Name That Postulate SAS ASA AAS SSA (when possible) Not enough info!

23. Name That Postulate (when possible) SSS AAA SSA Not enough info! SSA HL

24. Name That Postulate (when possible) SSA AAA Not enough info! HL SSA Not enough info!

25. Vertical Angles, Reflexive Sides and Angles When two triangles touch, there may be additional congruent parts. Vertical Angles Reflexive Side side shared by two triangles

26. Name That Postulate (when possible) SAS AAS SAS Reflexive Property Vertical Angles Reflexive Property SSA Not enough info!

27. When two triangles overlap, there may be additional congruent parts. Reflexive Side side shared by two triangles Reflexive Angle angle shared by two triangles Reflexive Sides and Angles

28. Let’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 A  F AC  FE

29. Try Some Proofs End Slide Show What’s Next

30. Choose a Problem. Problem #1 Problem #2 Problem #3 End Slide Show SSS SAS ASA

31. Problem #1

32. Step 1: Mark the Given

33. Reflexive Sides Vertical Angles Step 2: Mark... … if they exist.

34. Step 3: Choose a Method SSS SAS ASA AAS HL

35. Step 4: List the Parts STATEMENTSREASONS … in the order of the Method S S S

36. Step 5: Fill in the Reasons (Why did you mark those parts?) STATEMENTSREASONS S S S

37. S S S Step 6: Is there more? STATEMENTSREASONS The “Prove” Statement is always last !

38. Choose a Problem. Problem #1 Problem #2 Problem #3 End Slide Show SSS SAS ASA

39. Problem #2

40. Step 1: Mark the Given

41. Reflexive Sides Vertical Angles … if they exist. Step 2: Mark...

42. Step 3: Choose a Method SSS SAS ASA AAS HL

43. Step 4: List the Parts … in the order of the Method STATEMENTSREASONS S A S

44. Step 5: Fill in the Reasons (Why did you mark those parts?) STATEMENTSREASONS S A S

45. Step 6: Is there more? STATEMENTSREASONS The “Prove” Statement is always last ! S A S

46. Choose a Problem. Problem #1 Problem #2 Problem #3 End Slide Show SSS SAS ASA

47. Problem #3

48. Step 1: Mark the Given

49. Reflexive Sides Vertical Angles … if they exist. Step 2: Mark...

50. Step 3: Choose a Method SSS SAS ASA AAS HL

51. Step 4: List the Parts … in the order of the Method STATEMENTSREASONS A S A

52. Step 5: Fill in the Reasons (Why did you mark those parts?) STATEMENTSREASONS A S A

53. Step 6: Is there more? STATEMENTSREASONS The “Prove” Statement is always last ! A S A

54. Choose a Problem. Problem #1 Problem #2 Problem #3 End Slide Show SSS SAS ASA

55. Choose a Problem. Problem #4 Problem #5 End Slide Show AAS HL

56. Problem #4 55 StatementsReasons AAS Given Vertical Angles Thm AAS Postulate

57. Choose a Problem. Problem #4 Problem #5 End Slide Show AAS HL

58. StatementsReasons 57 HL Given Reflexive Property HL Postulate Problem #5 1.  ABC,  ADC right  s Given  ABC,  ADC right  s, Prove:

59. Congruence Proofs 1. Mark the Given. 2. Mark … Reflexive Sides or Angles / Vertical Angles Also: mark info implied by given info. 3. Choose a Method. (SSS, SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more?

60. Given implies Congruent Parts midpoint parallel segment bisector angle bisector perpendicular segmentsanglessegments angles

61. Example Problem

62. Step 1: Mark the Given … and what it implies

63. Reflexive Sides Vertical Angles Step 2: Mark... … if they exist.

64. Step 3: Choose a Method SSS SAS ASA AAS HL

65. Step 4: List the Parts STATEMENTSREASONS … in the order of the Method S A S

66. Step 5: Fill in the Reasons (Why did you mark those parts?) STATEMENTSREASONS S A S

67. S A S Step 6: Is there more? STATEMENTSREASONS 1. 2. 3. 4. 5. 1. 2. 3. 4. 5.

68. Midpoint implies segments. Back STATEMENTSREASONS S … 3. 3. Given

69. Parallel implies angles. Back STATEMENTSREASONS A A

70. Seg. bisector implies segments. Back STATEMENTSREASONS S S …

71. Angle bisector implies angles. Back STATEMENTSREASONS A …

72. implies right ( ) angles. Back STATEMENTSREASONS A … S 4. Given 4.

73. Congruent Triangles Proofs 1. Mark the Given and what it implies. 2. Mark … Reflexive Sides / Vertical Angles 3. Choose a Method. (SSS, SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more?

74. Using CPCTC in Proofs According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent. This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles. This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.

75. Corresponding Parts of Congruent Triangles For example, can you prove that sides AD and BC are congruent in the figure at right? The sides will be congruent if triangle ADM is congruent to triangle BCM. –Angles A and B are congruent because they are marked. –Sides MA and MB are congruent because they are marked. –Angles 1 and 2 are congruent because they are vertical angles. –So triangle ADM is congruent to triangle BCM by ASA. This means sides AD and BC are congruent by CPCTC.

76. StatementReason MA @ MBGiven ÐA @ ÐBGiven Ð1 @ Ð2Vertical angles DADM @ DBCMASA AD @ BCCPCTC Corresponding Parts of Congruent Triangles A two column proof that sides AD and BC are congruent in the figure at right is shown below:

77. StatementReason MA @ MBGiven ÐA @ ÐBGiven Ð1 @ Ð2Vertical angles DADM @ DBCMASA AD @ BCCPCTC Corresponding Parts of Congruent Triangles A two column proof that sides AD and BC are congruent in the figure at right is shown below:

78. StatementReason FR @ FOGiven RU @ OUGiven UF @ UF reflexive prop. DFRU @ DFOUSSS ÐR @ ÐOCPCTC Corresponding Parts of Congruent Triangles Sometimes it is necessary to add an auxiliary line in order to complete a proof For example, to prove Ð R @ Ð O in this picture

79. StatementReason FR @ FOGiven RU @ OUGiven UF @ UFSame segment DFRU @ DFOUSSS ÐR @ ÐOCPCTC Corresponding Parts of Congruent Triangles Sometimes it is necessary to add an auxiliary line in order to complete a proof For example, to prove Ð R @ Ð O in this picture