G.6

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Proving Triangles Congruent

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Two geometric figures with exactly the same size and shape.

The Idea of CongruenceThe Idea of Congruence

A C

B

DE

F

2

How much do you How much do you need to know. . .need to know. . .

. . . about two triangles to prove that they are congruent?

3

Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.

Corresponding PartsCorresponding Parts

ABC DEF

B

A C

E

D

F

1. AB DE

2. BC EF

3. AC DF

4. A D

5. B E

6. C F

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Do you need Do you need all six ?all six ?

NO !

SSSSASASAAASHL

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Side-Side-Side (SSS)Side-Side-Side (SSS)

1. AB DE

2. BC EF

3. AC DF

ABC DEF

B

A

C

E

D

F

Side

Side

Side

The triangles are congruent

by SSS.

If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

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The angle between two sides

Included AngleIncluded Angle

HGI G

GIH I

GHI H

This combo is called side-angle-side, or just SAS.

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Name the included angle:

YE and ES

ES and YS

YS and YE

Included AngleIncluded Angle

SY

E

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)Side-Angle-Side (SAS)

1. AB DE

2. A D

3. AC DF

ABC DEF

B

A

C

E

D

F

included

angle Side

Angle

Side

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.

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The side between two angles

Included SideIncluded Side

GI HI GH

This combo is called angle-side-angle, or just ASA.

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Name the included side:

Y and E

E and S

S and Y

Included SideIncluded Side

SY

E

YE

ES

SY

The other two sides are the

NON-INCLUDED sides.

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Angle-Side-Angle-Side-AngleAngle (ASA) (ASA)

1. A D

2. AB DE

3. B E

ABC DEF

B

A

C

E

D

F

included

side

Angle

Side

Angle

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.

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E

D

F

Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)

1. A D

2. B E

3. BC EF

ABC DEF

Non-included

side

B

AC

SideAngle

Angle

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.

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Warning:Warning: No SSA Postulate No SSA Postulate

There is no such thing as an SSA

postulate!

The triangles are NOTcongruent!

Side

Side

Angle

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Warning:Warning: No SSA Postulate No SSA Postulate

NOT CONGRUENT!

There is no such thing as an SSA

postulate!

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BUT:BUT: SSA DOES work in one SSA DOES work in onesituation!situation!

If we know that the two triangles

are right triangles!

Side

Side

Side

Angle

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We call thisWe call this

These triangles ARE CONGRUENT by HL!

HL,for “Hypotenuse – Leg”

Hypotenuse

Leg

Hypotenuse

RIGHT Triangles!

Remember! The

triangles must be RIGHT!

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Hypotenuse-Leg (HL)Hypotenuse-Leg (HL)

1.AB HL

2.CB GL

3. C and G are rt. ‘s

ABC DEF

The triangles

are congruent

by HL.

Right Triangle

LegHypotenuse

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.

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Warning:Warning: No AAA Postulate No AAA Postulate

A C

B

D

E

F

There is no such thing as an AAA

postulate!

NOT CONGRUENT!

Same Shapes

!

Different Sizes!

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Congruence Postulates Congruence Postulates and Theoremsand Theorems

• SSS• SAS• ASA• AAS• AAA?• SSA?• HL

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Name That PostulateName That Postulate

SASSASASAASA

AASAASSSASSA

(when possible)

Not enough info!

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Name That PostulateName That Postulate(when possible)

SSSSSSAAAAAA

SSASSA

Not enough info!

Not enough info!

SSASSAHL

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Name That PostulateName That Postulate(when possible)

SSASSA

AAAAAA

Not enough info!

Not enough info!

HL

SSASSA

Not enough info!

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Vertical Angles, Vertical Angles, Reflexive Sides and AnglesReflexive 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 PostulateName That Postulate(when possible)

SASASS

AASAAS

SASASS

Reflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SSSS

AANot enough

info!

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When two triangles overlap, there may be additional congruent parts.

Reflexive Sideside shared by two

triangles

Reflexive Angleangle shared by two

triangles

Reflexive Sides and AnglesReflexive Sides and Angles26

Let’s PracticeLet’s PracticeIndicate the additional information needed to enable us to apply the specified congruence postulate.

For ASA:

For SAS:

B D

For AAS: A F

AC FE

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Try Some Proofs

End Slide Show

What’s Next

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Choose a Problem.

Problem #1

Problem #2

Problem #3

End Slide Show

D

A B

C

E

C

D

AB

Z

W Y

X

SSS

SAS

ASA

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Problem #4

Statements Reasons

AAS

Given

Given

Vertical Angles Thm

AAS Postulate

Given: A C BE BDProve: ABE CBD

E

C

D

AB

4. ABE CBD

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Problem #5

3. AC AC

Statements Reasons

CB D

AHL

Given

Given

Reflexive Property

HL Postulate4. ABC ADC

1. ABC, ADC right s

AB AD

Given ABC, ADC right s,

Prove:

AB AD

ABC ADC

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Congruence Proofs1. Mark the Given.2. Mark …

Reflexive Sides or Angles / Vertical AnglesAlso: 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 Partsmidpoint

parallel

segment bisector

angle bisector

perpendicular

segments

angles

segments

angles

angles

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Example Problem

CB D

AGiven: AC bisects BAD AB ADProve: ABC ADC

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Step 1: Mark the Given

… and what it implies

CB D

AGiven: AC bisects BAD AB ADProve: ABC ADC

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•Reflexive Sides•Vertical Angles

Step 2: Mark . . .

… if they exist.

CB D

AGiven: AC bisects BAD AB ADProve: ABC ADC

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Step 3: Choose a Method

SSSSASASAAASHL

CB D

AGiven: AC bisects BAD AB ADProve: ABC ADC

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Step 4: List the Parts

STATEMENTS REASONS

… in the order of the Method

CB D

AGiven: AC bisects BAD AB ADProve: ABC ADC

BAC DAC

AB AD

AC AC

S

AS

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Step 5: Fill in the Reasons

(Why did you mark those parts?)

STATEMENTS REASONS

CB D

AGiven: AC bisects BAD AB ADProve: ABC ADC

BAC DAC

AB AD

AC AC

Given

Def. of Bisector

Reflexive (prop.)

S

AS

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S

AS

Step 6: Is there more?

STATEMENTS REASONS

CB D

AGiven: AC bisects BAD AB ADProve: ABC ADC

BAC DAC

AB AD

AC AC

Given

AC bisects BAD Given

Def. of Bisector

Reflexive (prop.)ABC ADC SAS (pos.)

1.2.3.4.5.

1.2.3.4.5.

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Congruent Triangles Proofs

1. Mark the Given and what it implies.2. Mark … Reflexive Sides / Vertical Angles3. 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 @ MB Given

ÐA @ ÐB Given

Ð1 @ Ð2 Vertical angles

DADM @ DBCM ASA

AD @ 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 @ MB Given

ÐA @ ÐB Given

Ð1 @ Ð2 Vertical angles

DADM @ DBCM ASA

AD @ 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 @ ÐO in this picture

Statement Reason

FR @ FO Given

RU @ OU Given

UF @ UF reflexive prop.

DFRU @ DFOU SSS

ÐR @ 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 @ ÐO in this picture

Statement Reason

FR @ FO Given

RU @ OU Given

UF @ UF Same segment

DFRU @ DFOU SSS

ÐR @ ÐO CPCTC

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