Proving Triangles Congruent

Proving Triangles Congruent

Two geometric figures with exactly the same size and shape.

The Idea of a CongruenceThe Idea of a Congruence

A C

B

DE

F

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

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

NO !

SSSSASHL

ASAAAS

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

1. AB DE

2. BC EF

3. AC DF

ABC DEF

C

B

A

E

D

F

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

The angle between two sides

Included AngleIncluded Angle

G I H

Name the included angle:

YE and ES

ES and YS

YS and YE

Included AngleIncluded Angle

SY

E

E

S

Y

Hypotenuse Leg (HL)Hypotenuse Leg (HL)

1. AB DE

2. B E

3. AC DF

ABC DEF

B

A

C

E

D

F

right angle

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

The side between two angles

Included SideIncluded Side

GI HI GH

Name the included side:

Y and E

E and S

S and Y

Included SideIncluded Side

SY

E

YE

ES

SY

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

1. A D

2. B E

3. BC EF

ABC DEF

B

A

C

E

D

F

Non-included

side

Warning:Warning: No SSA or ASS Postulate No SSA or ASS Postulate

A C

B

D

E

F

NOT CONGRUENT

There is no such thing as an SSA

or ASS postulate!

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

The Congruence PostulatesThe Congruence Postulates SSS

SAS

HL

ASA

AAS

SSA or ASS

AAA

Name That PostulateName That Postulate

SASSASASAASA

SSSSSSSSASSA

(when possible)

Name That PostulateName That Postulate(when possible)

ASAASA

SASASS

AAAAAA

SSASSA

Name That PostulateName That Postulate(when possible)

SASASS

SASSAS

SASASS

Reflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SSSS

AA

Name That PostulateName That Postulate(when possible)

Reflexive Property SSSSSS

none

SSSSAA

AAAAAA

(when possible)Name That PostulateName That Postulate

Vertical Angles

ASAASAReflexive Property SSSS

AA

SASASS

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

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

For ASA:

For SAS:

For AAS:

N V

MK TU

M U