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

2. 1 1 2 2 1 1 2 2

3. DO NOW!

4. DO NOW!

5. The Idea of Congruence
Prior Knowledge
Two geometric figures with exactly the same size and shape. The pre-image and image of an isometry are congruent. A C B D E F

6. Prior Knowledge A M N \\ \
\\\ \
\\\ B \\ C L

7. I DO A M N B C L

8. I DO A M N B C L

9. I DO A M N B C L

10. I DO A M N \ \ B C L

11. I DO A M N \\ B \\ C L

12. I DO A M N
\\\
\\\ B C L

13. Write the congruency statement
Partners
Write the congruency statement

14. Write the congruency statement
Partners
Write the congruency statement

15. ON YOUR OWN
Module 1- Lesson 20
p. 116
Example 1

16. M1-L21 P. 116

17. M1-L21 P. 119

18. M1-L21 P. 120

19. INDEPENDENT PRACTICE
Homework
Module 1- L21 pp

20. INDEPENDENT PRACTICE

21. INDEPENDENT PRACTICE

22. INDEPENDENT PRACTICE

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

24. 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
ABC   DEF E D F

25. NO ! Do you need all six ? 5 ways to prove triangles congruent SSS SAS
ASA
AAS HL

26. Wikki Stix Triangles Cut the following lengths 5 cm 6cm 8 cm
Use these lengths to form a triangle on your clear sheet.
You must connect end to end and no bending.
Compare your triangle to your partners.
How are they alike?
How are they different?

27. The triangles are congruent by SSS.
Side-Side-Side (SSS)
If the sides of one triangle are congruent to the corresponding 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.

28. With a Partner

31. On your own

32. Module 1- Lesson24
p. 139 #2

34. Wikki Stix Triangles On your paper,
Now cut a wikki stik for the 3rd side
to complete the triangle.
Compare your triangle with your partners.
Are they congruent?

35. side-angle-side, or just SAS.
Included Angle
The angle formed by the two referenced sides.
GIH I
GHI H
HGI G
This combo is called
side-angle-side, or just SAS.

36. 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.

37. SAS
Theorem

38. 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.

39. Mark any additional information you can justify.
I DO
Mark any additional information you can justify.
Then determine if the two triangles can be proven congruent
by the SAS Theorem. If yes, write the congruency statement.

40. Mark any additional information you can justify.
I DO
Mark any additional information you can justify.
Then determine if the two triangles can be proven congruent
by the SAS Theorem. If yes, write the congruency statement.

41. Can these two triangles be proven congruent
With a Partner
Can these two triangles be proven congruent
using the Side-Angle-Side Theorem?
If so, write the congruency statement.

42. Mark any additional information you can justify.
With a Partner
Mark any additional information you can justify.
Then determine if the two triangles can be proven congruent
by the SAS Theorem. If yes, write the congruency statement.

43. Mark any additional information you can justify.
ON YOUR OWN
Mark any additional information you can justify.
Then determine if the two triangles can be proven congruent
by the SAS Theorem. If yes, write the congruency statement.

44. NOT CONGRUENT Angle is not included ON YOUR OWN
Mark any additional information you can justify.
Then determine if the two triangles can be proven congruent
by the SAS Theorem. If yes, write the congruency statement.
NOT CONGRUENT
Angle is not included

45. Module 1-Lesson 24 pp
#1-10

56. EXIT SLIP
If the triangles are congruent, state the theorem that
justifies the congruency and write the congruency statement. 2. 1. 4. 3.

60. 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.

61. 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.

62. angle-side-angle, or just ASA.
Included Side
The side which connects the two referenced angles. GI GH HI
This combo is called
angle-side-angle, or just ASA.

63. 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.

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

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

66. 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

67. 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!

68. 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   HLG
The triangles are congruent by HL.

69. 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!

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

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

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

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

74. 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

75. 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!

76. 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

77. 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:

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

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

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

81. 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

82. 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?

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

84. Example Problem

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

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

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

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

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

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

91. 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?

92. 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.

93. 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.

94. 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

95. 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

96. 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

97. 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