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Proving Triangles Congruent

Proving Triangles Congruent. F. B. A. C. E. D. The Idea of a Congruence. Two geometric figures with exactly the same size and shape. How much do you need to know. . . . . . about two triangles to prove that they are congruent?.

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Proving Triangles Congruent

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  1. Proving Triangles Congruent

  2. F B A C E D The Idea of a Congruence Two geometric figures with exactly the same size and shape.

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

  4. Corresponding Parts • AB DE • BC EF • AC DF •  A  D •  B  E •  C  F B A C E F D If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. ABC DEF

  5. SSS SAS ASA RHS Do you need all six ? NO ! S means side & A means angle

  6. 1) Side-Side-Side (SSS) E B F A D C • AB DE • BC EF • AC DF ABC DEF

  7. 2) Side-Angle-Side (SAS) B E F A C D • AB DE • A D • AC DF ABC DEF included angle

  8. Included Angle The angle between two sides H G I

  9. E Y S Included Angle Name the included angle: YE and ES ES and YS YS and YE E S Y

  10. 3) Angle-Side-Angle(ASA) B E F A C D • A D • AB  DE • B E ABC DEF included side

  11. Included Side The side between two angles GI GH HI

  12. E Y S Included Side Name the includedside : Y and E E and S S and Y YE ES SY

  13. Angle-Angle-Side (AAS) B E F A C D • A D • B E • AC EF NOT  Non-included side

  14. Angle-Angle-Side (AAS) Sometimes will be ( ASA ) B E * F A * C D • A D • B E • BC  EF C F ( why ?) ABC DEF

  15. 4) Right angle- hypotenuse - side (RHS) F B C A D E ∘ • A =D = 90 • AB  DE • BC  EF ABC DEF

  16. Warning: No SSA Postulate There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT

  17. Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C F D NOT CONGRUENT

  18. SSS correspondence • ASA correspondence • SAS correspondence • RHS correspondence • SSA correspondence • AAA correspondence The Congruence Postulates

  19. Example In the given figure prove that : AB ≅ AD In ∆∆ ADC , ABC we have : 1) DC = BC ( given ) A 2) ∡ DCA = ∡BCA ( given ) 3) AC = AC ( common side ) • • ∴ ∆ ACD ≅ ∆ ACB ( SAS postulate) C ∴ AB ≅ AD ( CPCTC ) Corresponding parts of ≅ ∆ s are ≅ فيهما :ADC , ABC ∆∆ B D 1) ( معطى ) DC = BC ∡ DCA = ∡BCA 2) ( معطى ) 3) ( ضلع مشترك ) AC = AC ∴ ∆ ACD ≅ ∆ ACB ( SAS مسلمة ) ∴ AB ≅ AD ( ( الأجزاء المتناظرة في المثلثات المتطابقة تكون متطابقة

  20. Name That Postulate (when possible) SAS ASA SSA SSS

  21. Name That Postulate (when possible) AAA ASA SSA SAS

  22. Name That Postulate (when possible) Vertical Angles Reflexive Property SAS SAS Reflexive Property Vertical Angles SSA SAS

  23. ACTIVITY : Name That Postulate (when possible)

  24. ACTIVITY: Name That Postulate (when possible)

  25. Let’s Practice ACFE Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AF For AAS ASA :

  26. ACTIVITY Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS ASA :

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