Proving Triangles Congruent

Two geometric figures with exactly the same size and shape.

The Idea of a Congruence

A C

B

DE

F

How much do you need to know. . .

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

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

Corresponding Parts

ABC DEF

B

A C

E

D

F

1. AB DE2. BC EF3. AC DF4. A D5. B E6. C F

Do you need all six ?

NO !

SSSSASASAAAS

Side-Side-Side (SSS)

1. AB DE2. BC EF3. AC DF

ABC DEF

B

A

C

E

D

F

If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent

Side-Angle-Side (SAS)

1. AB DE2. A D3. AC DF

ABC DEF

B

A

C

E

D

F

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent

included side

The angle between two sidesIncluded Angle

G I H

Name the included angle:

YE and ES ES and YS YS and YE

Included Angle

SY

E

E S Y

Angle-Side-Angle (ASA)

1. A D2. AB DE3. B E

ABC DEF

B

A

C

E

D

F

included side

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

The side between two anglesIncluded Side

GI HI GH

Name the included angle:

Y and E E and S S and Y

Included Side

SY

E

YEESSY

Angle-Angle-Side (AAS)

1. A D2. B E3. BC EF

ABC DEF

B

A

C

E

D

F

Non-included

sideIf 2 angles and a non-included side of 1 triangle are congruent to 2 angles and the corresponding non-included side of another triangle, then the 2 triangles are congruent

Warning: No SSA Postulate

A C

B

D

E

F

NOT CONGRUENT

There is no such thing as an SSA

postulate!

Warning: No AAA Postulate

A C

B

D

E

F

There is no such thing as an AAA

postulate!

NOT CONGRUENT

The Congruence Postulates SSS

correspondence ASA

correspondence SAS

correspondence AAS

correspondence SSA

correspondence AAA

correspondence

Name That Postulate

SAS ASA

SSSSSA

(when possible)

Name That Postulate(when possible)

ASA

SAS

AAA

SSA

Name That Postulate(when possible)

SAS

SAS

SAS

Reflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SS

A

HW: Name That Postulate(when possible)

(when possible)HW: Name That Postulate

Let’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

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

For ASA:

For SAS:

For AAS: