1. Warm-Up #38
Line M goes through the points (7, -1) and (-2, 3). Write an equation for a line perpendicular to M and through the origin.
What are the new points of A(-3, 3) and B(2, 5) if you want to use the rule (x, y)  (x -3, y + 4)? What type of transformation is this?

2. Homework
SSS-SAS-ASA-AAS Congruence page 1&2

3. Triangle Congruence

4. congruent polygons:
are polygons with congruent corresponding parts - their matching sides and angles B Y A X C Z D W
Polygon ABCD  Polygon XYZW

5. The Right Match
Congruent Not congruent A D F E C B

6. corresponding parts of congruent triangles are congruent
pg. 203
CPCTC:
corresponding parts of congruent triangles are congruent

7. Corresponding sides and anglesT A B S R
Corresponding Angles
Corresponding sides
AB RS
BC ST
AC RT

8. Like congruence of segments and angles, congruence of triangles is reflexive, symmetric, and transitive.

9. Proving Vertical Angle Theorem
Vertical Angles Theorem
Vertical angles are congruent
1 3, 2 4

10. Proving Vertical Angle Theorem
GIVEN
5 and 6 are a linear pair,
6 and 7 are a linear pair
PROVE
5 7
Statements
Reasons 1
5 and 6 are a linear pair, Given
6 and 7 are a linear pair 2
5 and 6 are supplementary, Linear Pair Postulate
6 and 7 are supplementary 3
Congruent Supplements Theorem

11. If  A   D and  B   E, then  C   F.
THEOREM
Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
If  A   D and  B   E, then  C   F.
Goal 1

12. We will use:
Use the SSS Postulate
Use the SAS Postulate
Use the HL Theorem
Use ASA Postulate
Use AAS Theorem
to prove that two triangles are congruent.

13. SSS: Side-Side-Side Postulate
If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. H Q G P F R
 GHF   PQR

14. Write a proof statement:

15. SAS: Side-Angle-Side Postulate
If two sides and included angle of one triangle are congruent to two sides and included angle of another triangle, then the two triangles are congruent. B D C F A E
 BCA   FDE

16. Write a proof statement:

17. ASA: Angle-Side-Angle Postulate
If two angles and included side of one triangle are congruent to two angles and included side of another triangle, then the two triangles are congruent. B P G K H N
 HGB   NKP

18. Write a proof statement:

19. AAS: Angle-Angle-Side Postulate
If two angles and nonincluded side of one triangle are congruent to two angles and nonincluded side of another triangle, then the two triangles are congruent. C T D G M X
 CDM   TGX
Chris Giovanello, LBUSD Math Curriculum Office, 2004

20. Chris Giovanello, LBUSD Math Curriculum Office, 2004
Write a proof statement:
Chris Giovanello, LBUSD Math Curriculum Office, 2004

21. HL Theorem Hypotenuse - Leg Congruent 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 .

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

23. 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

24. 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.

25. 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.

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

27. 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.

28. 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

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

30. 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

31. 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

32. 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

33. 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

34. Theroem 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 .

35. 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

36. 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.

37. 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