Proving Δ s are  : SSS, SAS, HL, ASA, AAS, & CPCTC

1. Proving Δs are  : SSS, SAS, HL, ASA, AAS, & CPCTC

2. SSSSide-Side-Side  Postulate • If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are .

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

4. 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

5. 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

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

7. 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

8. SASSide-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 .

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

10. 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

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

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

13. 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 right s 4. Right s Theorem 5. SAS Postulate D R G A

14. HLHypotenuse - 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 .

15. ASAAngle-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.

16. 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.

17. 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

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

19. 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.

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

21. 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.

22. 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.

23. 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

24. CPCTC

25. CPCTC Given: Prove: D  B STATEMENTS REASONS