1. Proving Δs are  : SSS, SAS, HL, ASA, & AAS
GEOMETRY
Ms. olifer

2. Objectives Use the SSS Postulate Use the SAS Postulate
Use the HL Theorem
Use ASA Postulate
Use AAS Theorem
Prove problems using triangle congruence theorems/postulates

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

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

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

6. Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent.
GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true. Explain your reasoning.
SOLUTION
Yes. The statement is true.
DFG HJK
Side DG HK, Side DF JH,and Side FG JK.
So by the SSS Congruence postulate, DFG HJK.

7. GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true. Explain your reasoning. 2.
ACB CAD
SOLUTION
BC AD
GIVEN :
PROVE :
ACB CAD
PROOF:
It is given that BC AD By Reflexive property
AC AC, But AB is not congruent CD.

8. GUIDED PRACTICE
for Example 1
Therefore the given statement is false and ABC is not
Congruent to CAD because corresponding sides
are not congruent

9. GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true. Explain your reasoning.
QPT RST 3.
SOLUTION
QT TR , PQ SR, PT TS
GIVEN :
PROVE :
QPT RST
PROOF:
It is given that QT TR, PQ SR, PT TS. So by
SSS congruence postulate, QPT RST. Yes the statement is true.

10. Postulate (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 .

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

12. EXAMPLE 2
Use the SAS Congruence Postulate
Write a proof.
GIVEN
BC DA, BC AD
PROVE
ABC CDA
STATEMENTS
REASONS
Given
BC DA S
Given
BC AD
BCA DAC
Alternate Interior Angles Theorem A
AC CA
Reflexive Property of Congruence S

13. EXAMPLE 2
Use the SAS Congruence Postulate
STATEMENTS
REASONS
ABC CDA
SAS Congruence Postulate

14. Given: RS  RQ and ST  QT Prove: Δ QRT  Δ SRT.
Example 3:
Given: RS  RQ and ST  QT Prove: Δ QRT  Δ SRT. S Q R T

15. S Q R
Example 3: T
Statements Reasons________ 1. RS  RQ; ST  QT 1. Given 2. RT  RT 2. Reflexive 3. Δ QRT  Δ SRT 3. SSS Postulate

16. Given: DR  AG and AR  GR Prove: Δ DRA  Δ DRG.
Example 4:
Given: DR  AG and AR  GR Prove: Δ DRA  Δ DRG. D R A G

17. Example 4:
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

18. 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.

19. 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 .

20. 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.

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

22. Example 5:
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.

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

24. Example 6:
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.

25. 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.

26. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons:
Given
If || lines, then alt. int. s are 
Vertical Angles Theorem
ASA Congruence Postulate

27. HOMEWORK
Textbook