G.6

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G.6
Proving Triangles Congruent
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The Idea of Congruence
Two geometric figures with exactly the same
size and shape.
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How much do you need to know. . .
. . . about two triangles to
prove that they are congruent?
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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
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Do you need all six ?
NO !
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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
E




Side

F
D

Side

1. AB ? DE
2. BC ? EF
3. AC ? DF

?ABC ? ? DEF
The triangles are congruent by SSS.
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Included Angle
The angle between two sides

• GHI
• H

• GIH
• I

• HGI
• G

This combo is called side-angle-side, or just
SAS.
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Included Angle
Name the included angle YE and ES ES and
YS YS and YE
? YES or ?E
? YSE or ?S
? EYS or ?Y
The other two angles are the NON-INCLUDED angles.
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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


B

E





Side

F
A



C
Side
D

1. AB ? DE
2. ?A ? ? D
3. AC ? DF

Angle
?ABC ? ? DEF
The triangles are congruent by SAS.
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Included Side
The side between two angles
GI
GH
HI
This combo is called angle-side-angle, or just
ASA.
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Included Side
Name the included side ?Y and ?E ?E and ?S
?S and ?Y
YE
ES
SY
The other two sides are the NON-INCLUDED sides.
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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


B

E
Angle





Side

F
A



C
D
Angle

1. ?A ? ? D
2. AB ? DE
3. ? B ? ? E

?ABC ? ? DEF
The triangles are congruent by ASA.
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Angle-Angle-Side (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

Angle


Side










Angle

1. ?A ? ? D
2. ? B ? ? E
3. BC ? EF

?ABC ? ? DEF
The triangles are congruent by AAS.
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Warning No SSA Postulate
There is no such thing as an SSA postulate!
Side
Angle
Side
The triangles are NOTcongruent!
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Warning No SSA Postulate
There is no such thing as an SSA postulate!
NOT CONGRUENT!
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BUT SSA DOES work in one situation!
If we know that the two triangles are right
triangles!
Side
Side
Side
Angle
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We call this
HL, for Hypotenuse Leg
Remember! The triangles must be RIGHT!
Hypotenuse
Hypotenuse
Leg
RIGHT Triangles!
These triangles ARE CONGRUENT by HL!
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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.
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Warning No AAA Postulate
There is no such thing as an AAA postulate!
Different Sizes!
E
Same Shapes!
B
A
C
F
D
NOT CONGRUENT!
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Congruence Postulates and Theorems

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

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Name That Postulate
(when possible)
SAS
ASA
SSA
AAS
Not enough info!
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Name That Postulate
(when possible)
AAA
Not enough info!
SSS
SSA
SSA
Not enough info!
HL
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Name That Postulate
(when possible)
Not enough info!
Not enough info!
SSA
SSA
HL
AAA
Not enough info!
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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
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Name That Postulate
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Reflexive Property
Vertical Angles
SSA
AAS
Not enough info!
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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
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Lets Practice
Indicate the additional information needed to
enable us to apply the specified congruence
postulate.
For ASA
?B ? ?D
For SAS
?A ? ?F
For AAS
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Whats Next
Try Some Proofs End Slide Show
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Choose a Problem.
End Slide Show
Problem 1
SSS
Problem 2
SAS
Problem 3
ASA
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Problem 1
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Step 1 Mark the Given
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Step 2 Mark . . .

• Reflexive Sides
• Vertical Angles

if they exist.
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Step 3 Choose a Method
SSS SAS ASA AAS HL
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Step 4 List the Parts
S
S
S
in the order of the Method
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Step 5 Fill in the Reasons
S
S
S
(Why did you mark those parts?)
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Step 6 Is there more?
The Prove Statement is always last !
S
S
S
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Choose a Problem.
End Slide Show
Problem 1
SSS
Problem 2
SAS
Problem 3
ASA
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Problem 2
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Step 1 Mark the Given
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Step 2 Mark . . .

• Reflexive Sides
• Vertical Angles

if they exist.
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Step 3 Choose a Method
SSS SAS ASA AAS HL
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Step 4 List the Parts
S
A
S
in the order of the Method
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Step 5 Fill in the Reasons
S
A
S
(Why did you mark those parts?)
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Step 6 Is there more?
The Prove Statement is always last !
S
A
S
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Choose a Problem.
End Slide Show
Problem 1
SSS
Problem 2
SAS
Problem 3
ASA
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Problem 3
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Step 1 Mark the Given
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Step 2 Mark . . .

• Reflexive Sides
• Vertical Angles

if they exist.
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Step 3 Choose a Method
SSS SAS ASA AAS HL
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Step 4 List the Parts
A
S
A
in the order of the Method
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Step 5 Fill in the Reasons
A
S
A
(Why did you mark those parts?)
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Step 6 Is there more?
The Prove Statement is always last !
A
S
A
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Choose a Problem.
End Slide Show
Problem 1
SSS
Problem 2
SAS
Problem 3
ASA
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Choose a Problem.
End Slide Show
Problem 4
AAS
Problem 5
HL
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Problem 4
AAS
Statements Reasons




Given
Vertical Angles Thm
Given
AAS Postulate
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Choose a Problem.
End Slide Show
Problem 4
AAS
Problem 5
HL
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Problem 5
HL
Given ?ABC, ?ADC right ?s, Prove
Statements Reasons




Given
1. ?ABC, ?ADC right ?s
Given
Reflexive Property
HL Postulate
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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?
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Given implies Congruent Parts
midpoint
parallel
segment bisector
angle bisector
perpendicular
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Example Problem
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Step 1 Mark the Given
and what it implies
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Step 2 Mark . . .

• Reflexive Sides
• Vertical Angles

if they exist.
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Step 3 Choose a Method
SSS SAS ASA AAS HL
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Step 4 List the Parts
in the order of the Method
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Step 5 Fill in the Reasons
S
A
S
(Why did you mark those parts?)
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Step 6 Is there more?
S
A
S
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Midpoint implies segments.
Back
S

3. Given
3.
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Parallel implies angles.
Back
A
A
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Seg. bisector implies segments.
Back
S

S
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Angle bisector implies angles.
Back
A

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implies right ( ) angles.
Back
A

S
4.
4. Given
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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?
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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.

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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.

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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
MA _at_ MB Given
ÐA _at_ ÐB Given
Ð1 _at_ Ð2 Vertical angles
DADM _at_ DBCM ASA
AD _at_ BC CPCTC
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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
MA _at_ MB Given
ÐA _at_ ÐB Given
Ð1 _at_ Ð2 Vertical angles
DADM _at_ DBCM ASA
AD _at_ BC CPCTC
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Corresponding Parts of Congruent Triangles

• Sometimes it is necessary to add an auxiliary
  line in order to complete a proof
• For example, to prove ÐR _at_ ÐO in this picture

Statement Reason
FR _at_ FO Given
RU _at_ OU Given
UF _at_ UF reflexive prop.
DFRU _at_ DFOU SSS
ÐR _at_ ÐO CPCTC
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Corresponding Parts of Congruent Triangles

• Sometimes it is necessary to add an auxiliary
  line in order to complete a proof
• For example, to prove ÐR _at_ ÐO in this picture

Statement Reason
FR _at_ FO Given
RU _at_ OU Given
UF _at_ UF Same segment
DFRU _at_ DFOU SSS
ÐR _at_ ÐO CPCTC