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Proving Triangles Congruent

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Proving Triangles Congruent

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F

B

A

C

E

D

The Idea of a Congruence

Two geometric figures with exactly the same size and shape.

How much do you

need to know. . .

. . . about two triangles

to prove that they

are congruent?

Corresponding Parts

- AB DE
- BC EF
- AC DF
- A D
- B E
- C F

B

A

C

E

F

D

In Lesson 4.2, you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.

ABC DEF

SSS

SAS

ASA

AAS

Do you need all six ?

NO !

Side-Side-Side (SSS)

E

B

F

A

D

C

- AB DE
- BC EF
- AC DF

ABC DEF

Side-Angle-Side (SAS)

B

E

F

A

C

D

- AB DE
- A D
- AC DF

ABC DEF

included

angle

Included Angle

The angle between two sides

H

G

I

E

Y

S

Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

E

S

Y

Angle-Side-Angle (ASA)

B

E

F

A

C

D

- A D
- AB DE
- B E

ABC DEF

included

side

Included Side

The side between two angles

GI

GH

HI

E

Y

S

Included Side

Name the included side:

Y and E

E and S

S and Y

YE

ES

SY

Angle-Angle-Side (AAS)

B

E

F

A

C

D

- A D
- B E
- BC EF

ABC DEF

Non-included

side

Warning: No SSA Postulate

There is no such thing as an SSA postulate!

E

B

F

A

C

D

NOT CONGRUENT

Warning: No AAA Postulate

There is no such thing as an AAA postulate!

E

B

A

C

F

D

NOT CONGRUENT

- SSS correspondence
- ASA correspondence
- SAS correspondence
- AAS correspondence
- SSA correspondence
- AAA correspondence

The Congruence Postulates

Name That Postulate

(when possible)

SAS

ASA

SSA

SSS

Name That Postulate

(when possible)

AAA

ASA

SSA

SAS

Name That Postulate

(when possible)

Vertical Angles

Reflexive Property

SAS

SAS

Reflexive Property

Vertical Angles

SSA

SAS

HW: Name That Postulate

(when possible)

HW: Name That Postulate

(when possible)

Let’s Practice

ACFE

Indicate the additional information needed to enable us to apply the specified congruence postulate.

For ASA:

B D

For SAS:

AF

For AAS:

HW

Indicate the additional information needed to enable us to apply the specified congruence postulate.

For ASA:

For SAS:

For AAS: