1. 4.3 and 4.4 Proving Δs are  : SSS and SAS AAS and ASA

2. Objectives:
Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem and the SAS and SSS.
Use congruence postulates and theorems in real-life problems.

3. Postulate 21: 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, then the triangles are congruent.

4. Meaning: A Side-Side-Side (SSS)
___
___
If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are . B C
___
Meaning: E
If seg AB  seg ED,
seg AC  seg EF
seg BC  seg DF,
then ΔABC  ΔEDF.
___
___ D
___ F

5. Theorem 4.5: Angle-Angle-Side (AAS) 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.

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

7. If seg BC  seg YX, seg AC  seg ZX, and C  X, then ΔABC  ΔZXY.) ( C A X Z

8. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem
Given: A  D, C  F, BC  EF Prove: ∆ABC  ∆DEF

9. Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

10. Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

11. Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. UZ ║WX AND UW║WX. 1 2 3 4

12. Ex. 2 Proving Triangles are Congruent
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.

13. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons: 1.

14. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons: 1. Given

15. Proof: Statements: Reasons: BD  BC Given AD ║ EC D  C ABD  EBC

16. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons:
Given
Alternate Interior Angles

17. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons:
Given
Alternate Interior Angles
Vertical Angles Theorem

18. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons:
Given
Alternate Interior Angles
Vertical Angles Theorem
ASA Congruence Theorem

19. Given: seg QR  seg UT, RS  TS, QS=10, US=10 Prove: ΔQRS  ΔUTS

20. Proof Statements Reasons 1. 1. given 2. QS=US 2. subst. prop. =
3. Seg QS  seg US Def of  segs.
4. Δ QRS  Δ UTS SSS post
seg QR  seg UT,
RS  TS, QS=10, US=10

21. Given: seg WX  seg. XY, seg VX  seg ZX, Prove: Δ VXW  Δ ZXY1 2 Y V

22. Proof Statements Reasons 1. seg WX  seg. XY 1. given seg. VX  seg ZX
2. 1   vert s thm
3. Δ VXW  Δ ZXY SAS post

23. Given: seg RS  seg RQ and seg ST  seg QT Prove: Δ QRT  Δ SRT.

24. Proof Statements Reasons 1. Seg RS  seg RQ 1. Given seg ST  seg QT
2. Seg RT  seg RT 2. Reflex prop 
3. Δ QRT  Δ SRT 3. SSS post

25. Given: seg DR  seg AG and seg AR  seg GR Prove: Δ DRA  Δ DRG.

26. Proof Statements seg DR  seg AG Seg AR  seg GR 2. seg DR  Seg DR
3.DRG & DRA are rt. s
4.DRG   DRA
5. Δ DRG  Δ DRA
Reasons
Given
reflex. Prop of 
 lines form 4 rt. s
4. Rt. s thm
5. SAS post.

27. Assignment
Workbook pages