Proving Triangles Congruent

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Why do we study triangles so much?

• Because they are the only “rigid” shape.

• Show pictures of triangles

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?

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

Corresponding PartsCorresponding Parts

ABC DEF

B

A C

E

D

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

1. AB DE2. BC EF3. AC DF

ABC DEF

Order matters!!

B

A

C

E

D

F

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

Distance Formula• The Cartesian coordinate system is

really two number lines that are perpendicular to each other.

• We can find distance between two points via………..

• Mr. Moss’s favorite formulaa2 + b2 = c2

Distance Formula• The distance formula is from

Pythagorean’s Theorem.• Remember distance on the number

line is the difference of the two numbers, so

• And• So

212

212 yyxxd

12 xxa 12 yyb

To find distance:• You can use the Pythagorean

theorem or the distance formula.• You can get the lengths of the legs

by subtracting or counting.• A good method is to align the points

vertically and then subtract them to get the distance between them

• Find the distance between (-3, 7) and (4, 3)

_____

4,73,47,3

65164947 22 d

• Do some examples via Geo Sketch

Class Work• Page 240, # 1 – 13 all

Warm Up• Are these triangle pairs congruent

by SSS? Give congruence statement or reason why not.

D

BA

C

E G

F

H

I K

J

N

MO

Q

P

R

L

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 sidesIncluded 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

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

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!

Why no SSA Postulate?• We would get in trouble if someone

spelled it backwards. (bummer!)• Really – SSA allows the possibility of

having two different solutions.• Remember: It only takes one counter

example to prove a conjecture wrong.• Show problem on Geo Sketch

* A Right Δ has one Right .

Right

Leg

Leg – one of the sides that form the right of the Δ. Shorter than the hyp.

Hypotenuse –- Longest side of a Rt.

Δ - Opp. of Right

Th(4-6) Hyp – Leg Thm. (HL)• If the hyp. & a leg of 1 right Δ are

to the hyp & a leg of another rt. Δ, then the Δ’s are .

Group Class Work• Do problem 18 on page 245

Class Work• Pg 244 # 1 – 17 all

Warm Up• Are these triangle pairs congruent?

State postulate or theorem and congruence statement or reason why not.

IA B

CD

R

S T

J

KM

L

G

E

F

H

U V

WX

N

O

Q

P Y

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

1. A D2.AB DE3. B E

ABC DEF

B

A

C

E

D

F

included side

The side between two anglesIncluded 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

Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)• The proof of this is based on ASA.• If you know two angles, you really

know all three angles.• The difference between ASA and

AAS is just the location of the angles and side.

• ASA – Side is between the angles• AAS –Side is not between the angles

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

Triangle Congruence Triangle Congruence (5 of them)(5 of them)

SSS postulate ASA postulate SAS postulate AAS theorem HL Theorem

Right Triangle CongruenceRight Triangle CongruenceWhen dealing with right triangles, you

will sometimes see the following definitions of congruence:

HL - Hypotenuse LegHA – Hypotenuse Angle

(AAS)LA - Leg Angle (AAS of ASA)LL – Leg Leg Theorem (SAS)

Name That PostulateName That Postulate

SASSAS ASAASA

SSSSSSSSASSA

(when possible)

Name That PostulateName That Postulate(when possible)

ASAASA

SASASS

AAAAAA

SSASSA

Name That PostulateName That Postulate(when possible)

SASASS

SASSAS

SASASS

Reflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SSSS

AA

Name That PostulateName That Postulate(when possible)

(when possible)Name That PostulateName 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

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

For ASA:

For SAS:

For AAS:

Class Work• Pg 251, # 1 – 17 all

CPCTC• Proving shapes are congruent proves ALL

corresponding dimensions are congruent. • For Triangles: Corresponding Parts of Congruent

Triangles are Congruent • This includes, but not limited to, corresponding

sides, angles, altitudes, medians, centroids, incenters, circumcenters, etc.

They are ALL CONGRUENT!

Warm UpGiven that triangles ABC and DEF are congruent, andG and H are the centroids, find AH.

AG = 7x + 8JH = 5x - 8Find DJ

JI

A

B

C F

E

D

G H