Proving Triangles Congruent - Mrs....

Visit www.worldofteaching.com For 100’s of free powerpoints.

Proving Triangles Congruent

1

Two geometric figures with exactly the same size and shape.

The Idea of Congruence

A C

B

DE

F

2

How much do you need to know. . . !

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

3

Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.

!

Corresponding Parts

ΔABC ≅ Δ DEF

B

A C

E

D

F

1. AB ≅ DE

2. BC ≅ EF

3. AC ≅ DF

4. ∠ A ≅ ∠ D

5. ∠ B ≅ ∠ E

6. ∠ C ≅ ∠ F

4

Do you need all six ?

NO !

SSS SAS ASA AAS

5

Side-Side-Side (SSS)

1. AB ≅ DE

2. BC ≅ EF

3. AC ≅ DF

ΔABC ≅ Δ DEF

B

A

C

E

D

F

Side

Side

Side

The triangles are congruent by SSS.

If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

6

The angle between two sides

Included Angle

∠ HGI ∠ G

∠ GIH ∠ I

∠ GHI ∠ H

This combo is called side-angle-side, or just SAS.

7

Name the included angle: !

YE and ES

ES and YS

YS and YE

Included Angle

SY

E

∠ YES or ∠E

∠ YSE or ∠S

∠ EYS or ∠Y The other two angles are the

NON-INCLUDED angles.

8

Side-Angle-Side (SAS)

1. AB ≅ DE

2. ∠A ≅ ∠ D

3. AC ≅ DF

ΔABC ≅ Δ DEF

B

AC

E

D

F

included

angle Side

Angle

Side

The triangles are congruent by

SAS.

If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.

9

The side between two angles

Included Side

GI HI GH

This combo is called angle-side-angle, or just ASA.

10

Name the included side: !

∠Y and ∠E

∠E and ∠S

∠S and ∠Y

Included Side

SY

E

YE

ES

SY

The other two sides are the NON-

INCLUDED sides.

11

Angle-Side-Angle (ASA)

1. ∠A ≅ ∠ D

2. AB ≅ DE

3. ∠ B ≅ ∠ E

ΔABC ≅ Δ DEF

B

AC

E

D

F

!included

sideAngle

Side

Angle

The triangles are congruent by

ASA.

If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.

12

E

D

F

Angle-Angle-Side (AAS)

1. ∠A ≅ ∠ D

2. ∠ B ≅ ∠ E

3. BC ≅ EF

ΔABC ≅ Δ DEF

Non-included side

B

AC

SideAngle

Angle

The triangles are congruent by

AAS.

If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent.

13

Warning: No SSA Postulate

There is no such thing as an SSA

postulate!

The triangles are NOTcongruent!

Side

Side

Angle

14

Warning: No AAA Postulate

A C

B

D

E

F

There is no such thing as an AAA

postulate!

NOT CONGRUENT!

Same Shapes!

Different Sizes!

15

Congruence Postulates and Theorems

• SSS • SAS • ASA • AAS • AAA? • SSA?

16

Name That Postulate

SAS ASA

AASSSA

(when possible)

Not enough info!

17

Name That Postulate(when possible)

SSSAAA

SSA

Not enough info!

Not enough info!

18

Name That Postulate(when possible)

SSA

AAA

Not enough info!

Not enough info!

SSA

Not enough info!

19

Vertical Angles, Reflexive Sides and Angles

When two triangles touch, there may be additional congruent parts.

Vertical Angles

!

Reflexive Side

side shared by two

triangles

20

Name That Postulate(when possible)

SAS

AAS

SASReflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SSA

Not enough info!

21

When two triangles overlap, there may be additional congruent parts.

Reflexive Side side shared by two

triangles

Reflexive Angle angle shared by two

triangles

Reflexive Sides and Angles22

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

23