1. 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
CPCTC Theorem

3. Vocabulary
Bisect: to cut into two equal parts

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

5. 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

6. 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

7. 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.

8. 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.

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

10. 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.

11. 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 .

12. 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

13. 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

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

15. 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

16. R 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

17. 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

18. 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

19. Theroem (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 .
Note: Right Triangles Only

20. Postulate (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.

21. Theorem (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.

22. 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

23. Theroem (CPCTC) Corresponding Parts of Congruent Triangles are Congruent
When two triangles are congruent, there are 6 facts that are true about the triangles:
the triangles have 3 sets of congruent (of equal length) sides and 
the triangles have 3 sets of congruent (of equal measure) angles.
Use this after you have shown that two figures are congruent. Then you could say that Corresponding parts of the two congruent figures are also congruent to each other.

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

25. 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.

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

27. 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.

28. 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.

29. Writing Proofs Proofs are used to prove what you are finding.
Geometric proofs can be written in one of two ways: two columns, or a paragraph. A paragraph proof is only a two-column proof written in sentences
List the given statements and then list the conclusion to be proved
Draw a figure and mark the figure accordingly along with your proofs

30. 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