Proving
Triangles
Congruent
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1
G.6

2
The Idea of Congruence
Two geometric figures with exactly the same size and shape.
F
B
A C E D

3
How much do you need to know. . .
. . . about two triangles to prove that they are congruent?

4
Corresponding Parts
Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.
1. AB  DE
2. BC  EF
3. AC  DF
4.
 A   D
5.
 B   E
6.
 C   F
 ABC   DEF

5
Do you need all six ?
NO !
SSS
SAS
ASA
AAS
HL

6
Side-Side-Side (SSS)
If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
Side
Side
1. AB  DE
2. BC  EF
3. AC  DF
Side
 ABC   DEF
The triangles are congruent by
SSS.

7
Included Angle
The angle between two sides
 H G I
 G
 G I H
 I
 G H I
 H
This combo is called side-angle-side, or just SAS.

Y S
E
8
Included Angle
Name the included angle:
Y E and E S
E S and Y S
Y S and Y E
The other two angles are the
NON-INCLUDED angles.
 Y E S or  E
 Y S E or  S
 E Y S or  Y

9
Side-Angle-Side (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
Side
1. AB  DE
2.
 A   D
3. AC  DF
Side
Angle
 ABC   DEF
The triangles are congruent by
SAS.

10
Included Side
The side between two angles
GI HI
This combo is called
GH angle-side-angle, or just ASA.

Y S
E
11
Included Side
Name the included side:



Y and
E and
S and



E
S
Y
YE
ES
SY
The other two sides are the
NON-INCLUDED sides.

Angle-Side-
Angle
(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
Angle
Side
Angle
1.
 A   D
2. AB  DE
3.
 B   E
 ABC   DEF
The triangles are congruent by
ASA.

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
Angle
Side
1.
 A   D
2.
 B   E
3. BC  EF
Angle
 ABC   DEF
The triangles are congruent by
AAS.

14
Warning: No SSA Postulate
There is no such thing as an SSA postulate!
Side
Angle
Side
The triangles are
NOTcongruent!

15
Warning: No SSA Postulate
There is no such thing as an SSA postulate!
NOT CONGRUENT!

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

17
We call this
HL, for “Hypotenuse – Leg”
Hypotenuse
Leg
Hypotenuse
Remember!
The triangles must be
RIGHT!
RIGHT Triangles!
These triangles ARE CONGRUENT by 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
1.AB
 HL
2.CB
 GL
3.
 C and  G are rt.  ‘s
 ABC   DEF
The triangles are congruent by HL.

19
Warning: No AAA Postulate
There is no such thing as an AAA postulate!
Same
Shapes!
B
A
E
C
D
NOT CONGRUENT!
Different
Sizes!
F

and Theorems
• SSS
• SAS
• ASA
• AAS
• AAA?
• SSA?
• HL

21
Name That Postulate
(when possible)
SAS
ASA
SSA
Not enough info!
AAS

22
Name That Postulate
(when possible)
AAA Not enough info!
SSS
SSA
Not enough info!
SSA
HL

23
Name That Postulate
(when possible)
Not enough info!
Not enough info!
SSA
SSA
HL
Not enough info!
AAA

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

25
Name That Postulate
(when possible)
Reflexive
Property
SAS
Vertical
Angles
SAS
Vertical
Angles
AAS
Reflexive
Property
SSA
Not enough info!

26
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 triangles

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

28
What’s Next
Try Some Proofs
End Slide Show

Choose a
Problem.
Problem #1
SSS
Problem #2
SAS
Problem #3
ASA
29
End Slide Show
B A
D
A
W
E
C
B
X
D
C
Y
Z

Problem #4
Given:  A   C
BE  BD
Prove: ABE  CBD
A
E
AAS
C
B
D
Statements Reasons
Given
Vertical Angles Thm
Given
4.
ABE  CBD
AAS Postulate
55

Problem #5
Given

ABC ,

ADC right
 s,
AB

AD
Prove:
3.
AC

AC 
ABC
 
ADC
B
Statements
1.

ABC ,

ADC right
 s
AB

AD
A
HL
C
Given
D
Reasons
Given
Reflexive Property
4.
ABC  ADC HL Postulate
57

58
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?

59
Given implies Congruent Parts midpoint
 segments parallel
segment bisector
angle bisector
perpendicular
 angles
 segments
 angles
 angles

60
Example Problem
Given: AC bisects  BAD
AB  AD
Prove: ABC  ADC
B
A
C D

61
Step 1: Mark the Given
Given: AC bisects  BAD
AB  AD
Prove: ABC  ADC
B
… and what it implies
A
C D

Step 2: Mark . . .
Given: AC bisects  BAD
AB  AD
Prove: ABC  ADC
62
• Reflexive Sides
• Vertical Angles
A
B C D
… if they exist.

63
Step 3: Choose a Method
Given: AC bisects  BAD
AB  AD
Prove: ABC  ADC
B
SSS
SAS
ASA
AAS
HL
A
C D

64
A
S
Step 4:
Given:
List the Parts
AC bisects  BAD
AB  AD
Prove: ABC  ADC
A
B
REASONS
C
S
STATEMENTS
AB  AD
 BAC   DAC
AC  AC
D
… in the order of the Method

65
A
S
Step 5: Fill in the Reasons
Given: AC bisects  BAD
AB  AD
Prove: ABC  ADC
B
S
STATEMENTS
AB  AD
REASONS
Given
A
C D
 BAC   DAC
AC  AC
Def. of Bisector
Reflexive (prop.)
(Why did you mark those parts?)

66
Step 6: Is there more?
Given: AC bisects  BAD
AB  AD
Prove: ABC  ADC
B
A
C
S
A
S
1.
2.
3.
4.
5.
STATEMENTS REASONS
AB  AD
1.
Given
AC bisects  BAD
2.
Given
 BAC   DAC 3.
Def. of Bisector
AC  AC
ABC  ADC
4.
5.
Reflexive (prop.)
SAS (pos.)
D

72
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?

73
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.

74
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.

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
Given MA @ MB
ÐA @ ÐB
Ð1 @ Ð2
DADM @
DBCM
AD @ BC
Given
Vertical angles
ASA
CPCTC

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
Given MA @ MB
ÐA @ ÐB
Ð1 @ Ð2
DADM @
DBCM
AD @ BC
Given
Vertical angles
ASA
CPCTC

Corresponding Parts of
Congruent Triangles


Sometimes it is necessary to add an auxiliary
line in order to complete a proof
For example, to prove Ð R @ Ð O in this picture
Statement
FR @ FO
RU @ OU
UF @ UF
Reason
Given
Given reflexive prop.
DFRU @ DFOU SSS
ÐR @ ÐO CPCTC

Corresponding Parts of
Congruent Triangles


Sometimes it is necessary to add an auxiliary
line in order to complete a proof
For example, to prove Ð R @ Ð O in this picture
Statement
FR @ FO
RU @ OU
UF @ UF
Reason
Given
Given
Same segment
DFRU @ DFOU SSS
ÐR @ ÐO CPCTC