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Proving Triangles Congruent

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Proving Triangles Congruent

F

B

A

C

E

D

What does it mean for 2 triangles to be congruent?

Corresponding angles and sides should have the same measure.

How much do you

need to know. . .

. . . about two triangles

to prove that they

are congruent?

Corresponding Parts

- AB DE
- BC EF
- AC DF
- A D
- B E
- C F

B

A

C

E

F

D

If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.

ABC DEF

SSS

SAS

ASA

AAS

Do you need all six ?

NO !

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)

E

B

F

A

D

C

- AB DE
- BC EF
- AC DF

ABC DEF

Side-Angle-Side (SAS)

SAS Congruence Conjecture

If 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)

B

E

F

A

C

D

- AB DE
- A D
- AC DF

ABC DEF

included

angle

Included Angle

The angle between two sides

G

E

Y

S

Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

E

S

Y

Included Side

The side between two angles

GH

E

Y

S

Included Side

Name the included angle:

Y and E

E and S

S and Y

YE

ES

SY

Angle-Side-Angle (ASA)

ASA Congruence Conjecture

If 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)

B

E

F

A

C

D

- A D
- AB DE
- B E

ABC DEF

included

side

Angle-Angle-Side (AAS or SAA)

AAS Congruence Conjecture

If 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)

B

E

F

A

C

D

- A D
- B E
- BC EF

ABC DEF

Non-included

side

Warning: No SSA Postulate

There is no such thing as an SSA postulate!

E

B

F

A

C

D

NOT CONGRUENT

Warning: No AAA Postulate

There is no such thing as an AAA postulate!

E

B

A

C

F

D

NOT CONGRUENT

- SSS correspondence
- ASA correspondence
- SAS correspondence
- AAS correspondence
- SSA correspondence
- AAA correspondence

The Congruence Postulates

Name That Postulate

(when possible)

SAS

ASA

SSA

SSS

Name That Postulate

(when possible)

AAA

ASA

SSA

SAS

Name That Postulate

(when possible)

Vertical Angles

Reflexive Property

SAS

SAS

Reflexive Property

Vertical Angles

SSA

SAS

HW: Name That Postulate

(when possible)

HW: Name That Postulate

(when possible)

Let’s Practice

ACFE

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

For ASA:

B D

For SAS:

AF

For AAS:

HW

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

For ASA:

For SAS:

For AAS:

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