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Lesson 4.4 - 4.5 Proving Triangles Congruent

Lesson 4.4 - 4.5 Proving Triangles Congruent

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Lesson 4.4 - 4.5 Proving Triangles Congruent

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  1. Lesson 4.4 - 4.5Proving Triangles Congruent

  2. Triangle Congruency Short-Cuts If you can prove one of the following short cuts, you have two congruent triangles • SSS (side-side-side) • SAS (side-angle-side) • ASA (angle-side-angle) • AAS (angle-angle-side) • HL (hypotenuse-leg) right triangles only!

  3. Built – In Information in Triangles

  4. Identify the ‘built-in’ part

  5. Shared side SSS Vertical angles SAS Parallel lines -> AIA Shared side SAS

  6. SOME REASONS For Indirect Information • Def of midpoint • Def of a bisector • Vert angles are congruent • Def of perpendicular bisector • Reflexive property (shared side) • Parallel lines ….. alt int angles • Property of Perpendicular Lines

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

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

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

  10. Angle-Angle-Side (AAS) B E F A C D • A D • B E • BC  EF ABC DEF Non-included side

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

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

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

  14. This is called a common side. It is a side for both triangles. We’ll use the reflexive property.

  15. HL( hypotenuse leg ) is used only with right triangles, BUT, not all right triangles. ASA HL

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

  17. Name That Postulate (when possible)

  18. Name That Postulate (when possible)

  19. Closure Question

  20. 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:

  21. G K I H J 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. Ex 4 ΔGIH ΔJIK by AAS

  22. B A C D E 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. Ex 5 ΔABC ΔEDC by ASA

  23. 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. Ex 6 E A C B D ΔACB ΔECD by SAS

  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. Ex 7 J K L M ΔJMK ΔLKM by SAS or ASA

  25. 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. J T Ex 8 L K V U Not possible

  26. SSS (Side-Side-Side) Congruence Postulate • If three sides of one triangle are congruent to three sides of a second triangle, the two triangles are congruent. If Side Side Side Then ∆ABC ≅ ∆PQR

  27. Example 1 Prove: ∆DEF ≅ ∆JKL From the diagram, SSS Congruence Postulate. ∆DEF ≅ ∆JKL

  28. SAS (Side-Angle-Side) Congruence Postulate • If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

  29. Angle-Side-Angle (ASA) Congruence Postulate • If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, the two triangles are congruent. ∠A ≅ ∠D If Angle Side Angle ∠C ≅ ∠F Then ∆ABC ≅ ∆DEF

  30. Example 2 Prove: ∆SYT ≅ ∆WYX

  31. Side-Side-Side Postulate • SSS postulate: If two triangles have three congruent sides, the triangles are congruent.

  32. Angle-Angle-Side Postulate • If two angles and a non included side are congruent to the two angles and a non included side of another triangle then the two triangles are congruent.

  33. Angle-Side-Angle Postulate • If two angles and the side between them are congruent to the other triangle then the two angles are congruent.

  34. Side-Angle-Side Postulate • If two sides and the adjacent angle between them are congruent to the other triangle then those triangles are congruent.

  35. Which Congruence Postulate to Use? 1. Decide whether enough information is given in the diagram to prove that triangle PQR is congruent to triangle PQS. If so give a two-column proof and state the congruence postulate.

  36. ASA • If 2 angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the 2 triangles are congruent. A Q S C R B

  37. AAS • If 2 angles and a nonincluded side of one triangle are congruent to 2 angles and the corresponding nonincluded side of a second triangle, then the 2 triangles are congruent. A Q S C R B

  38. AAS Proof • If 2 angles are congruent, so is the 3rd • Third Angle Theorem • Now QR is an included side, so ASA. A Q S C R B

  39. Example • Is it possible to prove these triangles are congruent?

  40. Example • Is it possible to prove these triangles are congruent? • Yes - vertical angles are congruent, so you have ASA

  41. Example • Is it possible to prove these triangles are congruent?

  42. Example • Is it possible to prove these triangles are congruent? • No. You can prove an additional side is congruent, but that only gives you SS

  43. Example • Is it possible to prove these triangles are congruent? 2 1 3 4

  44. Example • Is it possible to prove these triangles are congruent? • Yes. The 2 pairs of parallel sides can be used to show Angle 1 =~ Angle 3 and Angle 2 =~ Angle 4. Because the included side is congruent to itself, you have ASA. 2 1 3 4

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

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

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

  48. E Y S Included Side Name the included side: Y and E E and S S and Y YE ES SY

  49. Side-Side-Side Congruence Postulate SSS Post. - If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If then,

  50. 1) Given 2) SSS Using SSS Congruence Post. Prove: • 1) • 2)