1. Proving Triangles are Congruent SSS, SAS; ASA; AAS
CCSS: G.CO7

2. Standards for Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.  
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

3. CCSS:G.CO 7
USE the definition of congruence in terms of rigid motions to SHOW that two triangles ARE congruent if and only if corresponding pairs of sides and corresponding pairs of angles ARE congruent.

4. ESSENTIAL QUESTION How do we show that triangles are congruent?
How do we use triangle congruence to plane and write proves ,and prove that constructions are valid?

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

6. Proving Triangles are Congruent:
SSS and SAS

7.  ABC  DEF SSS AND SAS CONGRUENCE POSTULATES
If all six pairs of corresponding parts (sides and angles) are
congruent, then the triangles are congruent.
Sides are
congruent
Angles are
congruent
Triangles are
congruent If
and
then
 ABC  DEF
1. AB DE
A D
2. BC EF
B E
3. AC DF
C F

8. S S S SSS AND SAS CONGRUENCE POSTULATES POSTULATE
POSTULATE 19 Side - Side - Side (SSS) Congruence Postulate
If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If
Side MN QR S
then  MNP  QRS
Side NP RS S
Side PM SQ S

9. SSS AND SAS CONGRUENCE POSTULATES
The SSS Congruence Postulate is a shortcut for proving two triangles are congruent without using all six pairs of corresponding parts.

10. S A S SSS AND SAS CONGRUENCE POSTULATES POSTULATE
POSTULATE 20 Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of a
second triangle, then the two triangles are congruent.
Side PQ WX If S
then  PQS WXY
Angle Q X A
Side QS XY S

11. Use the SSS Congruence Postulate to show that  ABC   FGH.
Congruent Triangles in a Coordinate Plane
Use the SSS Congruence Postulate to show that  ABC   FGH.
SOLUTION
AC = 3 and FH = 3
AC  FH
AB = 5 and FG = 5
AB  FG

12. Use the distance formula to find lengths BC and GH.
Congruent Triangles in a Coordinate Plane
Use the distance formula to find lengths BC and GH.
d = (x 2 – x1 ) 2 + ( y2 – y1 ) 2
d = (x 2 – x1 ) 2 + ( y2 – y1 ) 2
BC = (– 4 – (– 7)) 2 + (5 – 0 ) 2
GH = (6 – 1) 2 + (5 – 2 ) 2 = = = =

13. All three pairs of corresponding sides are congruent,
Congruent Triangles in a Coordinate Plane
BC = and GH = 34
BC  GH
All three pairs of corresponding sides are congruent,
 ABC   FGH by the SSS Congruence Postulate.

14. SSS  postulate
SAS  postulate

15. T C S G
The vertex of the included angle is the point in common.
SSS  postulate
SAS  postulate

16. SSS  postulate
Not enough info

17. SSS  postulate
SAS  postulate

18. Not Enough Info
SAS  postulate

19. SSS  postulate
Not Enough Info

20. SAS  postulate
SAS  postulate

22. Use the SSS Congruence Postulate to show that  NMP   DEF.
Congruent Triangles in a Coordinate Plane
Use the SSS Congruence Postulate to show that  NMP   DEF.
SOLUTION
MN = 4 and DE = 4
MN  DE
PM = 5 and FE = 5
PM  FE

23. Use the distance formula to find lengths PN and FD.
Congruent Triangles in a Coordinate Plane
Use the distance formula to find lengths PN and FD.
d = (x 2 – x1 ) 2 + ( y2 – y1 ) 2
d = (x 2 – x1 ) 2 + ( y2 – y1 ) 2
PN = (– 1 – (– 5)) 2 + (6 – 1 ) 2
FD = (2 – 6) 2 + (6 – 1 ) 2 =
= (-4) = =

24. All three pairs of corresponding sides are congruent,
Congruent Triangles in a Coordinate Plane
PN = and FD = 41
PN  FD
All three pairs of corresponding sides are congruent,
 NMP   DEF by the SSS Congruence Postulate.

25. Proving Triangles are Congruent
ASA; AAS

26. 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.

27. 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.

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

29. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem
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.

30. 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.

31. Ex. 1 Developing Proof
A. 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. You can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

32. 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.

33. Ex. 1 Developing Proof
B. 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.

34. 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

35. Ex. 1 Developing Proof
The two pairs of parallel sides can be used to show 1  3 and 2  4. Because the included side WZ is congruent to itself, ∆WUZ  ∆ZXW by the ASA Congruence Postulate. 1 2 3 4

36. 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.

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

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

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

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

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

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

43. Note:
You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D  C and A  E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.