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

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Proving Triangles Congruent

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Proving triangles congruent

Proving Triangles Congruent


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.


Proving triangles congruent

How much do you

need to know. . .

. . . about two triangles

to prove that they

are congruent?


Proving triangles 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


Proving triangles congruent

SSS

SAS

ASA

AAS

Do you need all six ?

NO !


Proving triangles congruent

But how many do you need?

And which ones?


Proving triangles congruent

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?


Proving triangles congruent

Does AAA work?

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


Proving triangles 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?


Proving 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.


Proving triangles congruent

Side-Side-Side (SSS)

E

B

F

A

D

C

  • AB DE

  • BC EF

  • AC DF

ABC DEF


Proving triangles congruent

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.


Proving triangles congruent

Side-Angle-Side (SAS)

B

E

F

A

C

D

  • AB DE

  • A D

  • AC DF

ABC DEF

included

angle


Proving triangles congruent

Included Angle

The angle between two sides

G


Proving triangles congruent

E

Y

S

Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

E

S

Y


Proving triangles congruent

Included Side

The side between two angles

GH


Proving triangles congruent

E

Y

S

Included Side

Name the included angle:

Y and E

E and S

S and Y

YE

ES

SY


Proving triangles congruent

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.


Proving triangles congruent

Angle-Side-Angle (ASA)

B

E

F

A

C

D

  • A D

  • AB  DE

  • B E

ABC DEF

included

side


Proving triangles congruent

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


Proving triangles congruent

Angle-Angle-Side (AAS)

B

E

F

A

C

D

  • A D

  • B E

  • BC  EF

ABC DEF

Non-included

side


Proving triangles congruent

Warning: No SSA Postulate

There is no such thing as an SSA postulate!

E

B

F

A

C

D

NOT CONGRUENT


Proving triangles congruent

Warning: No AAA Postulate

There is no such thing as an AAA postulate!

E

B

A

C

F

D

NOT CONGRUENT


Proving triangles congruent

  • SSS correspondence

  • ASA correspondence

  • SAS correspondence

  • AAS correspondence

  • SSA correspondence

  • AAA correspondence

The Congruence Postulates


Proving triangles congruent

Name That Postulate

(when possible)

SAS

ASA

SSA

SSS


Proving triangles congruent

Name That Postulate

(when possible)

AAA

ASA

SSA

SAS


Proving triangles congruent

Name That Postulate

(when possible)

Vertical Angles

Reflexive Property

SAS

SAS

Reflexive Property

Vertical Angles

SSA

SAS


Proving triangles congruent

HW: Name That Postulate

(when possible)


Proving triangles congruent

HW: Name That Postulate

(when possible)


Proving triangles congruent

Let’s Practice

ACFE

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

For ASA:

B D

For SAS:

AF

For AAS:


Proving triangles congruent

HW

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

For ASA:

For SAS:

For AAS:


Proving triangles congruent

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