Geometrycongruence opt

Proving Triangles
Congruent

The Idea of a Congruence

Two geometric figures with
exactly the same size and
shape.
F

B

A C E D

How much do you
need to know. . .

. . . about two triangles

to prove that they
are congruent?

Corresponding Parts
In Lesson 4.2, you learned that if all
six pairs of corresponding parts (sides
and angles) are congruent, then the
triangles are congruent.
B
• AB ≅ DE
• BC ≅ EF A
C
• AC ≅ DF
∀ ∠ A ≅ ∠ D ∆ABC ≅ ∆
DEF
∀ ∠ B ≅ ∠ E
E
∀ ∠ C ≅ ∠ F F
D

Do you need all six ?

NO !

SSS
SAS
ASA
AAS

Side-Side-Side (SSS)
B
E

A F
C D

• AB ≅ DE
• BC ≅ EF ∆ABC ≅ ∆ DEF
• AC ≅ DF

Side-Angle-Side (SAS)
B
E

A F
C D

• AB ≅ DE
• ∠A ≅ ∠ D ∆ABC ≅ ∆ DEF
• AC ≅ DF
included
angle

Included Angle
The angle between two sides

∠G ∠I ∠H

Included Angle

Name the included angle:
E

YE and ES ∠E
ES and YS ∠S

Y S YS and YE ∠Y

Angle-Side-Angle (ASA)
B
E

A F
C D

• ∠A ≅ ∠ D
• AB ≅ DE ∆ABC ≅ ∆ DEF
• ∠B ≅ ∠E
include
d
side

Included Side
The side between two angles

GI HI GH

Included Side

Name the included angle:
E

∠Y and ∠E YE
∠E and ∠S ES
Y S ∠S and ∠Y SY

Angle-Angle-Side (AAS)
B
E

A F
C D

• ∠A ≅ ∠ D
• ∠B ≅ ∠E ∆ABC ≅ ∆ DEF
• BC ≅ EF
Non-included
side

Warning: No SSA Postulate

There is no such
thing as an SSA
postulate!

B E

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

The Congruence Postulates
 SSS correspondence
 ASA correspondence
 SAS correspondence
 AAS correspondence
 SSA correspondence
 AAA correspondence

Name That Postulate
(when possible)

SAS
ASA

SSA SSS

Name That Postulate
(when possible)

AAA
ASA

SAS SSA

Name That Postulate
(when possible)

Vertical
Angles
Reflexive
Property SAS SAS

Vertical Reflexive
Angles SAS Property SSA

HW: Name That Postulate
(when possible)

Let’s Practice
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.

For ASA: ∠B ≅ ∠D
For SAS: AC ≅ FE
For AAS: ∠A ≅ ∠F

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

For ASA:

For SAS:

For AAS:

Geometrycongruence opt

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