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?

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 !

SSSSASASARHS

S means side &

A means angle

1) Side-Side-Side (SSS)

1. AB DE2. BC EF3. AC DF

ABC DEF

B

A

C

E

D

F

2) Side-Angle-Side (SAS)

1. AB DE2. A D3. AC DF

ABC DEF

B

A

C

E

D

F

included angle

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

3) Angle-Side-Angle (ASA)

1. A D2. AB DE3. B E

ABC DEF

B

A

C

E

D

F

included

side

The side between two anglesIncluded Side

GI HI GH

Name the included side :

Y and E E and S S and Y

Included Side

SY

E

YEESSY

Angle-Angle-Side (AAS)

1. A D2. B E3. AC EF

NOT

B

A

C

E

D

F

Non-included

side

Angle-Angle-Side (AAS)Sometimes will be ( ASA )

1. A D2. B E3. BC EF ABC

DEF

B

A

C

E

D

F

C F ( why ?)

* *

4) Right angle - hypotenuse - side (RHS)

1. A = D = 90

2. AB DE3. BC EF

ABC DEF

B

A C

E D

F

∘

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 RHS

correspondence SSA

correspondence AAA

correspondence

Example In the given figure prove that : AB ≅ AD

••B

CD

A

In ∆∆ ADC , ABC we have :

∆∆ ADC , ABC : فيهما

1) DC = BC ( given )

1) ( معطى )

DC = BC

2) ∡ DCA = ∡BCA ( given )

2) ( معطى )

∡ DCA = ∡BCA

3) AC = AC ( common side )

ضلع ) (3( مشترك

AC = AC

∴ ∆ ACD ≅ ∆ ACB ( SAS postulate)

∴ ∆ ACD ≅ ∆ ACB ( SAS مسلمة )

∴ AB ≅ AD ( CPCTC ) Corresponding parts of ≅ ∆ s are ≅

∴ AB ≅ AD ( ( المثلثات في المتناظرة األجزاءمتطابقة تكون المتطابقة

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

ACTIVITY : Name That Postulate(when possible)

(when possible)ACTIVITY : 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 ASA : A F AC FE

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

For ASA:

For SAS:

For AAS ASA :