Proving Triangles Congruent

Corresponding angles and sides should have the same measure.

What does it mean for 2 triangles to be congruent?

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 !

SSSSASASAAAS

But how many do you need?

And which ones?

What if we have all three angles?Use your protractor to draw Triangle ABC., so that:

• Angle A measures 70º, • Angle B measures 30º and

• Angle C measures 80º.

Now measure the lengths of the sides of your triangle.

Compare your results with your partner. Are your triangles congruent?

Does AAA work?

No, the angles will be the same, but the triangles don’t have to be congruent.

Will 3 sides be enough for congruence?

Make a triangle DEF so that:

• DE is 5 cm• EF is 8 cm

• DF is 12 cm

Now measure the angles. Compare your results with your partner. Are your triangles congruent?

Side-Side-Side (SSS)

SSS Congruence Conjecture

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

Side-Side-Side (SSS)

1. AB DE2. BC EF3. AC DF

ABC DEF

B

A

C

E

D

F

Side-Angle-Side (SAS)

SAS Congruence ConjectureIf 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle 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

included angle

The angle between two sidesIncluded Angle

G

Name the included angle:

YE and ES ES and YS YS and YE

Included Angle

SY

E

E S Y

The side between two anglesIncluded Side

GH

Name the included angle:

Y and E E and S S and Y

Included Side

SY

E

YEESSY

Angle-Side-Angle (ASA)

ASA Congruence ConjectureIf 2 angles and the included side of one triangle is congruent to two angles and the included side of another triangle, then the triangles are congruent.

Angle-Side-Angle (ASA)

1. A D2. AB DE3. B E

ABC DEF

B

A

C

E

D

F

included side

Angle-Angle-Side (AAS or SAA)

AAS Congruence ConjectureIf 2 angles and a non-included side of one triangle is congruent to the corresponding angles and side of another triangle, then the triangles are included

Angle-Angle-Side (AAS)

1. A D2. B E3. BC EF

ABC DEF

B

A

C

E

D

F

Non-included

side

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

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

For ASA:

For SAS:

For AAS:

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