1. Success Criteria LT: Today’s Agenda
I will write formal proofs showing how triangles are congruent when two corresponding angles and the included side are congruent.
Think of three things you can do to help insure your success in this class
I can …
I can…
Success Criteria
Today’s Agenda
Do Now
Lesson
HW#1
I can write formal proofs showing how triangles are congruent when two corresponding angles and the included side are congruent.

2. Second Semester Changes
Quiz corrections alone will no longer bump you up to a 3
Must have the homework related to the quiz turned in order to get your grade bumped up to a 3
Otherwise quiz corrections will only get you bumped up one point

3. Come to class ready to work-personal business stays outside
Come in, sit down and get to work
Phones must remain out of sight
Set your phone on shuffle and put it in your pocket
No talking during the lesson and private reasoning time until prompted
No bathroom breaks during the lesson
Wait until the lesson is over
10 10 still applies
Respect others
Please don’t put people down-even if you are joking
Respect that others are trying to learn-don’t disrupt

4. G.6 Proving Triangles Congruent Visit www.worldofteaching.com
For 100’s of free powerpoints.

5. The Idea of Congruence
Two geometric figures with exactly the same size and shape. A C B D E F

6. How much do you
need to know. . .
. . . about two triangles
to prove that they
are congruent?

7. Corresponding Parts
Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. B A C
AB  DE
BC  EF
AC  DF
 A   D
 B   E
 C   F E D F
ABC   DEF

8. Do you need all six ?
NO !
SSS
SAS
ASA
AAS HL

9. The triangles are congruent by SSS.
Side-Side-Side (SSS)
If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. B A C
Side E
Side F D
Side
AB  DE
BC  EF
AC  DF
ABC   DEF
The triangles are congruent by SSS.

10. side-angle-side, or just SAS.
Included Angle
The angle between two sides
GIH I
GHI H
HGI G
This combo is called
side-angle-side, or just SAS.

11. The other two angles are the NON-INCLUDED angles.
Name the included angle:
YE and ES
ES and YS
YS and YE S Y E
 YES or E
 YSE or S
 EYS or Y
The other two angles are the NON-INCLUDED angles.

12. Side-Angle-Side (SAS) The triangles are congruent by SAS.
If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.
included
angle B E
Side F A C
Side D
AB  DE
A   D
AC  DF
Angle
ABC   DEF
The triangles are congruent by SAS.

13. Warning: No SSA Postulate
There is no such thing as an SSA postulate!
Side
Angle
Side
The triangles are NOTcongruent!

14. There is no such thing as an SSA postulate!
Warning: No SSA Postulate
There is no such thing as an SSA postulate!
NOT CONGRUENT!

15. If we know that the two triangles are right triangles!
BUT: SSA DOES work in one
situation!
If we know that the two triangles are right triangles!
Side
Side
Side
Angle

16. These triangles ARE CONGRUENT by HL!
We call this
HL,
for “Hypotenuse – Leg”
Remember! The triangles must be RIGHT!
Hypotenuse
Hypotenuse
Leg
RIGHT Triangles!
These triangles ARE CONGRUENT by HL!

17. The triangles are congruent by HL.
Hypotenuse-Leg (HL)
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
Right Triangle
Leg
Hypotenuse
AB  HL
CB  GL
C and G are rt.  ‘s
ABC   DEF
The triangles are congruent by HL.

18. Reflexive Sides and Angles
Vertical Angles,
Reflexive Sides and Angles
When two triangles touch, there may be additional congruent parts.
Vertical Angles
Reflexive Side
side shared by two
triangles

19. Name That Postulate SAS SAS SSA AAS Vertical Angles Reflexive Property
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Vertical Angles
Reflexive Property
SSA
AAS
Not enough info!

20. Reflexive Sides and Angles
When two triangles overlap, there may be additional congruent parts.
Reflexive Side
side shared by two
triangles
Reflexive Angle
angle shared by two

21. DAY 2

22. Success Criteria LT: Today’s Agenda
I will write formal proofs showing how triangles are congruent when two corresponding angles and the included side are congruent.
Success Criteria
Today’s Agenda
Do Now
Lesson
HW#2
I can write formal proofs showing how triangles are congruent when two corresponding angles and the included side are congruent.

23. angle-side-angle, or just ASA.
Included Side
The side between two angles GI GH HI
This combo is called
angle-side-angle, or just ASA.

24. The other two sides are the NON-INCLUDED sides.
Name the included side:
Y and E
E and S
S and Y S Y E YE ES SY
The other two sides are the NON-INCLUDED sides.

25. Angle-Side-Angle (ASA) The triangles are congruent by ASA.
If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.
included
side B E
Angle
Side F A C D
Angle
A   D
AB  DE
 B   E
ABC   DEF
The triangles are congruent by ASA.

26. Angle-Angle-Side (AAS) The triangles are congruent by AAS.
If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent.
Non-included
side B A C
Angle E D F
Side
Angle
A   D
 B   E
BC  EF
ABC   DEF
The triangles are congruent by AAS.

27. There is no such thing as an AAA postulate!
Warning: No AAA Postulate
There is no such thing as an AAA postulate!
Different Sizes!
Same Shapes! E B A C F D
NOT CONGRUENT!

28. Congruence Postulates
and Theorems
SSS
SAS
ASA
AAS
AAA?
SSA? HL

29. Name That Postulate
(when possible)
SAS
ASA
SSA
AAS
Not enough info!

30. Name That Postulate AAA SSS SSA SSA HL (when possible)
Not enough info!
SSS
SSA
SSA
Not enough info! HL

31. Name That Postulate SSA SSA AAA HL (when possible) Not enough info!

32. Let’s Practice B  D AC  FE A  F
Indicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
A  F
For AAS:

33. Try Some Proofs End Slide Show
What’s Next
Try Some Proofs
End Slide Show

34. Choose a Problem. Problem #1 SSS Problem #2 SAS Problem #3 ASA
End Slide Show
Problem #1
SSS
Problem #2
SAS
Problem #3
ASA

35. AAS Problem #4 Statements Reasons Given Given AAS Postulate
Vertical Angles Thm
Given
AAS Postulate

36. HL Problem #5 Statements Reasons Given Given Reflexive Property
Given ABC, ADC right s,
Prove:
Statements
Reasons
Given
1. ABC, ADC right s
Given
Reflexive Property
HL Postulate

37. Congruence Proofs 1. Mark the Given. 2. Mark …
Reflexive Sides or Angles / Vertical Angles
Also: mark info implied by given info.
3. Choose a Method. (SSS , SAS, ASA)
4. List the Parts …
in the order of the method.
5. Fill in the Reasons …
why you marked the parts.
6. Is there more?

38. Given implies Congruent Parts
segments
midpoint
angles
parallel
segments
segment bisector
angles
angle bisector
angles
perpendicular

39. Example Problem

40. … and what it implies
Step 1: Mark the Given

41. Reflexive Sides
Vertical Angles
Step 2: Mark . . .
… if they exist.

42. Step 3: Choose a Method
SSS
SAS
ASA
AAS HL

43. Step 4: List the Parts S A … in the order of the Method STATEMENTS
REASONS S A
… in the order of the Method

44. Step 5: Fill in the Reasons
STATEMENTS
REASONS S A S
(Why did you mark those parts?)

45. Step 6: Is there more?
STATEMENTS
REASONS S 1. 2. 3. 4. 5. A S

46. Congruent Triangles Proofs
1. Mark the Given and what it implies.
2. Mark … Reflexive Sides / Vertical Angles
3. Choose a Method. (SSS , SAS, ASA)
4. List the Parts …
in the order of the method.
5. Fill in the Reasons …
why you marked the parts.
6. Is there more?

47. Using CPCTC in Proofs
According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent.
This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles.
This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.

48. Corresponding Parts of Congruent Triangles
For example, can you prove that sides AD and BC are congruent in the figure at right?
The sides will be congruent if triangle ADM is congruent to triangle BCM.
Angles A and B are congruent because they are marked.
Sides MA and MB are congruent because they are marked.
Angles 1 and 2 are congruent because they are vertical angles.
So triangle ADM is congruent to triangle BCM by ASA.
This means sides AD and BC are congruent by CPCTC.

49. Corresponding Parts of Congruent Triangles
A two column proof that sides AD and BC are congruent in the figure at right is shown below:
Statement
Reason MB
Given ÐB Ð2
Vertical angles
DBCM
ASA BC
CPCTC

50. Corresponding Parts of Congruent Triangles
A two column proof that sides AD and BC are congruent in the figure at right is shown below:
Statement
Reason MB
Given ÐB Ð2
Vertical angles
DBCM
ASA BC
CPCTC

51. Corresponding Parts of Congruent Triangles
Sometimes it is necessary to add an auxiliary line in order to complete a proof
For example, to prove ÐO in this picture
Statement
Reason FO
Given OU UF
reflexive prop.
DFOU
SSS ÐO
CPCTC

52. Corresponding Parts of Congruent Triangles
Sometimes it is necessary to add an auxiliary line in order to complete a proof
For example, to prove ÐO in this picture
Statement
Reason FO
Given OU UF
Same segment
DFOU
SSS ÐO
CPCTC