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4.3 to 4.5 Proving Δ s are  : SSS, SAS, HL, ASA, & AAS

4.3 to 4.5 Proving Δ s are  : SSS, SAS, HL, ASA, & AAS. OBJECTIVES. Use the SSS Postulate Use the SAS Postulate Use the HL Theorem Use ASA Postulate Use AAS Theorem. Postulate 19 (SSS) Side-Side-Side  Postulate.

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4.3 to 4.5 Proving Δ s are  : SSS, SAS, HL, ASA, & AAS

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  1. 4.3 to 4.5 Proving Δs are  : SSS, SAS, HL, ASA, & AAS

  2. OBJECTIVES • Use the SSS Postulate • Use the SAS Postulate • Use the HL Theorem • Use ASA Postulate • Use AAS Theorem

  3. Postulate 19 (SSS)Side-Side-Side  Postulate • If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are .

  4. E A F C D B More on the SSS Postulate If AB ED, AC EF, & BC DF, then ΔABC ΔEDF.

  5. Write a proof. GIVEN KL NL,KM NM PROVE KLMNLM Proof KL NL andKM NM It is given that LM LN. By the Reflexive Property, So, by the SSS Congruence Postulate, KLMNLM EXAMPLE 1: Use the SSS Congruence Postulate

  6. ACBCAD 1. GIVEN : BC AD ACBCAD PROVE : It is given that BC AD By Reflexive property AC AC, But AB is not congruent CD. PROOF: YOUR TURN: GUIDED PRACTICE Decide whether the congruence statement is true. Explain your reasoning. SOLUTION

  7. YOUR TURN (continued): GUIDED PRACTICE Therefore the given statement is false and ABC is not Congruent to CAD because corresponding sides are not congruent

  8. 2. QPTRST GIVEN : QT TR , PQ SR, PT TS PROVE : QPTRST It is given that QT TR, PQ SR, PT TS.So by SSS congruence postulate, QPT RST. Yes, the statement is true. PROOF: YOUR TURN: GUIDED PRACTICE Decide whether the congruence statement is true. Explain your reasoning. SOLUTION

  9. Postulate 20 (SAS)Side-Angle-Side  Postulate • If 2 sides and the included  of one Δ are  to 2 sides and the included  of another Δ, then the 2 Δs are .

  10. More on the SAS Postulate • If BC YX, AC ZX, & C X, then ΔABC  ΔZXY. B Y ) ( A C X Z

  11. BC DA,BC AD ABCCDA STATEMENTS REASONS S BC DA Given Given BC AD BCADAC A Alternate Interior Angles Theorem S ACCA Reflexive Property of Congruence EXAMPLE 2 Example 2: Use the SAS Congruence Postulate Write a proof. GIVEN PROVE

  12. EXAMPLE 2 Example 2 (continued): STATEMENTS REASONS ABCCDA SAS Congruence Postulate

  13. Given: DR  AG and AR GRProve: Δ DRA  ΔDRG. Example 4: D R A G

  14. Example 4 (continued): Statements_______ 1. DR  AG; AR  GR 2. DR  DR 3.DRG & DRA are rt. s 4.DRG   DRA 5. Δ DRG  Δ DRA Reasons____________ 1. Given 2. Reflexive Property 3.  lines form 4 rt. s 4. Right s Theorem 5. SAS Postulate D R G A

  15. Theorem 4.5 (HL)Hypotenuse - Leg  Theorem • If the hypotenuse and a leg of a right Δ are  to the hypotenuse and a leg of a second Δ, then the 2 Δs are .

  16. Postulate 21(ASA):Angle-Side-Angle 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, then the triangles are congruent.

  17. Theorem 4.6 (AAS): Angle-Angle-Side Congruence Theorem • If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.

  18. Proof of the Angle-Angle-Side (AAS) Congruence Theorem Given:A  D, C  F, BC  EF Prove: ∆ABC  ∆DEF D A B F C Paragraph Proof You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF. E

  19. Example 5: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

  20. Example 5 (continued): In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. Thus, you can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

  21. Example 6: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

  22. Example 6 (continued): In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

  23. Example 7: Given: AD║EC, BD  BC Prove: ∆ABD  ∆EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC. Use the fact that AD ║EC to identify a pair of congruent angles.

  24. Example 7 (continued): Reasons: • Given • Given • If || lines, then alt. int. s are  • Vertical Angles Theorem • ASA Congruence Postulate Statements: • BD  BC • AD ║ EC • D  C • ABD  EBC • ∆ABD  ∆EBC

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