Proving Triangles Congruent

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Two geometric figures with exactly the same size and shape.

The Idea of a CongruenceThe Idea of a Congruence

A C

B

DE

F

How much do you How much do you need to know. . .need to know. . .

. . . about two triangles to prove that they are congruent?

In Lesson 4.2, you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.

Corresponding PartsCorresponding Parts

∆ABC ≅ ∆ DEF

B

A C

ED

F

1. AB ≅ DE2. BC ≅ EF3. AC ≅ DF4. ∠ A ≅ ∠ D5. ∠ B ≅ ∠ E6. ∠ C ≅ ∠ F

Do you need Do you need all six ?all six ?

NO !

SSSSASASAAAS

Side-Side-Side (SSS)Side-Side-Side (SSS)

1. AB ≅ DE2. BC ≅ EF3. AC ≅ DF

∆ABC ≅ ∆ DEF

B

A

C

E

D

F

Side-Angle-Side (SAS)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 sides

Included AngleIncluded Angle

∠ G ∠ I ∠ H

Name the included angle:

YE and ES

ES and YS

YS and YE

Included AngleIncluded Angle

SY

E

∠ E∠ S∠ Y

Angle-Side-Angle-Side-AngleAngle (ASA) (ASA)

1. ∠A ≅ ∠ D2. AB ≅ DE3. ∠ B ≅ ∠ E

∆ABC ≅ ∆ DEF

B

A

C

E

D

F

included

side

The side between two angles

Included SideIncluded Side

GI HI GH

Name the included angle:

∠Y and ∠E

∠E and ∠S

∠S and ∠Y

Included SideIncluded Side

SY

E

YEESSY

Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)

1. ∠A ≅ ∠ D2. ∠ B ≅ ∠ E3. BC ≅ EF

∆ABC ≅ ∆ DEF

B

A

C

E

D

F

Non-included side

Warning:Warning: No SSA Postulate No SSA Postulate

A C

B

D

E

F

NOT CONGRUENT

There is no such thing as an SSA

postulate!

Warning:Warning: No AAA Postulate No AAA Postulate

A C

B

D

E

F

There is no such thing as an AAA

postulate!

NOT CONGRUENT

The Congruence PostulatesThe Congruence Postulates

SS S co rr es po nd e nc e

AS A co rre s po nd e nc e

SA S co rr es po nd e nc e

AA S co rre s po nd e nc e

SS A co rr es po nd e nc e

AA A co rre s po nd e nc e

Name That PostulateName That Postulate

SASSAS ASAASA

SSSSSSSSASSA

(when possible)

Name That PostulateName That Postulate(when possible)

ASAASA

SASSAS

AAAAAA

SSASSA

Name That PostulateName That Postulate(when possible)

SASSAS

SASSAS

SASSASReflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SSASSA

HW: Name That PostulateHW: Name That Postulate(when possible)

(when possible)HW: Name That PostulateHW: Name That Postulate

Let’s PracticeLet’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

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

For ASA:

For SAS:

For AAS: