5-5 & 5-6 SSS, SAS, ASA, & AAS

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 !

SSSSASASAAAS

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-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

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 Postulate No SSA Postulate

A C

B

D

E

F

NOT CONGRUENT

There is no such thing as an SSA

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 correspondence

ASA correspondence

SAS correspondence

AAS correspondence

SSA correspondence

AAA correspondence

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

HW: Name That PostulateHW: Name That Postulate(when possible)

(when possible)HW: Name That PostulateHW: Name That Postulate

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

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

For ASA:

For SAS:

For AAS:

Write a congruence statement for each pair of triangles represented.

A

B

FC

E

D

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Before we start…let’s get a few things straight

INCLUDED SIDE

A B

C

X Z

Y

Angle-Side-Angle (ASA) Congruence Postulate

Two angles and the INCLUDED side

Angle-Angle-Side (AAS) Congruence Postulate

Two Angles and One Side that is NOT included

} Your Only Ways To Prove Triangles Are

Congruent

Overlapping sides are congruent in

each triangle by the REFLEXIVE property

Vertical Angles

are congruen

t

Alt Int Angles are congruent

given parallel

lines

Things you can mark on a triangle when they aren’t marked.

Ex 1

statement. congruence a Write.

and ,, and In

LE

NLDENDΔLMNΔDEF

DEF NLM

Ex 2

What other pair of angles needs to be marked so that the two triangles are congruent by AAS?

F

D

E

M

L

N

NE

Ex 3

What other pair of angles needs to be marked so that the two triangles are congruent by ASA?

F

D

E

M

L

N

LD

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

ΔGIH ΔJIK by AAS

G

I

H J

KEx 4

ΔABC ΔEDC by ASA

B A

C

ED

Ex 5

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

ΔACB ΔECD by SAS

B

A

C

E

D

Ex 6

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

ΔJMK ΔLKM by SAS or ASA

J K

LM

Ex 7

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

Not possible

K

J

L

T

U

Ex 8

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

V

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(over Lesson 5-5)

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(over Lesson 5-5)

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