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Lesson 4.4 - 4.5 Proving Triangles Congruent. Triangle Congruency Short-Cuts. If you can prove one of the following short cuts, you have two congruent triangles SSS (side-side-side) SAS (side-angle-side) ASA (angle-side-angle) AAS (angle-angle-side) HL (hypotenuse-leg) right triangles only!.

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Lesson 4 4 4 5 proving triangles congruent

Lesson 4.4 - 4.5Proving Triangles Congruent


Triangle congruency short cuts
Triangle Congruency Short-Cuts

If you can prove one of the following short cuts, you have two congruent triangles

  • SSS (side-side-side)

  • SAS (side-angle-side)

  • ASA (angle-side-angle)

  • AAS (angle-angle-side)

  • HL (hypotenuse-leg) right triangles only!




Shared side

SSS

Vertical angles

SAS

Parallel lines

-> AIA

Shared side

SAS


Some reasons for indirect information
SOME REASONS For Indirect Information

  • Def of midpoint

  • Def of a bisector

  • Vert angles are congruent

  • Def of perpendicular bisector

  • Reflexive property (shared side)

  • Parallel lines ….. alt int angles

  • Property of Perpendicular Lines


Side-Side-Side (SSS)

E

B

F

A

D

C

  • AB DE

  • BC EF

  • AC DF

ABC DEF


Side-Angle-Side (SAS)

B

E

F

A

C

D

  • AB DE

  • A D

  • AC DF

ABC DEF

included

angle


Angle-Side-Angle (ASA)

B

E

F

A

C

D

  • A D

  • AB  DE

  • B E

ABC DEF

included

side


Angle-Angle-Side (AAS)

B

E

F

A

C

D

  • A D

  • B E

  • BC  EF

ABC DEF

Non-included

side


Warning: No AAA Postulate

There is no such thing as an AAA postulate!

E

B

A

C

F

D

NOT CONGRUENT


Warning: No SSA Postulate

There is no such thing as an SSA postulate!

E

B

F

A

C

D

NOT CONGRUENT


Name That Postulate

(when possible)

SAS

ASA

SSA

SSS


This is called a common side.

It is a side for both triangles.

We’ll use the reflexive property.


HL( hypotenuse leg ) is used

only with right triangles, BUT,

not all right triangles.

ASA

HL


Name That Postulate

(when possible)

Vertical Angles

Reflexive Property

SAS

SAS

Reflexive Property

Vertical Angles

SSA

SAS


Name That Postulate

(when possible)


Name That Postulate

(when possible)



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:


G

K

I

H

J

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

Ex 4

ΔGIH ΔJIK by AAS


B

A

C

D

E

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

Ex 5

ΔABC ΔEDC by ASA


Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

Ex 6

E

A

C

B

D

ΔACB ΔECD by SAS


Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

Ex 7

J

K

L

M

ΔJMK ΔLKM by SAS or ASA


Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

J

T

Ex 8

L

K

V

U

Not possible


Sss side side side congruence postulate
SSS (Side-Side-Side) Congruence Postulate SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • If three sides of one triangle are congruent to three sides of a second triangle, the two triangles are congruent.

If Side

Side

Side

Then

∆ABC ≅ ∆PQR


Example 1
Example 1 SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Prove: ∆DEF ≅ ∆JKL

From the diagram,

SSS Congruence Postulate.

∆DEF ≅ ∆JKL


Sas side angle side congruence postulate
SAS (Side-Angle-Side) Congruence Postulate SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

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


Angle-Side-Angle (ASA) Congruence Postulate SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, the two triangles are congruent.

∠A ≅ ∠D

If Angle

Side

Angle

∠C ≅ ∠F

Then

∆ABC ≅ ∆DEF


Example 2
Example 2 SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Prove: ∆SYT ≅ ∆WYX


Side side side postulate
Side-Side-Side Postulate SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • SSS postulate: If two triangles have three congruent sides, the triangles are congruent.


Angle angle side postulate
Angle-Angle-Side Postulate SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • If two angles and a non included side are congruent to the two angles and a non included side of another triangle then the two triangles are congruent.


Angle side angle postulate
Angle-Side-Angle Postulate SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • If two angles and the side between them are congruent to the other triangle then the two angles are congruent.


Side angle side postulate
Side-Angle-Side Postulate SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • If two sides and the adjacent angle between them are congruent to the other triangle then those triangles are congruent.


Which congruence postulate to use
Which Congruence Postulate to Use? SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

1. Decide whether enough information is given in the diagram to prove that triangle PQR is congruent to triangle PQS. If so give a two-column proof and state the congruence postulate.


ASA SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • If 2 angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the 2 triangles are congruent.

A

Q

S

C

R

B


AAS SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • If 2 angles and a nonincluded side of one triangle are congruent to 2 angles and the corresponding nonincluded side of a second triangle, then the 2 triangles are congruent.

A

Q

S

C

R

B


Aas proof
AAS Proof SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • If 2 angles are congruent, so is the 3rd

  • Third Angle Theorem

  • Now QR is an included side, so ASA.

A

Q

S

C

R

B


Example
Example SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • Is it possible to prove these triangles are congruent?


Example1
Example SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • Is it possible to prove these triangles are congruent?

  • Yes - vertical angles are congruent, so you have ASA


Example2
Example SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • Is it possible to prove these triangles are congruent?


Example3
Example SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • Is it possible to prove these triangles are congruent?

  • No. You can prove an additional side is congruent, but that only gives you SS


Example4
Example SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • Is it possible to prove these triangles are congruent?

2

1

3

4


Example5
Example SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • Is it possible to prove these triangles are congruent?

  • Yes. The 2 pairs of parallel sides can be used to show Angle 1 =~ Angle 3 and Angle 2 =~ Angle 4. Because the included side is congruent to itself, you have ASA.

2

1

3

4


Included Angle SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

The angle between two sides

H

G

I


E SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Y

S

Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

E

S

Y


Included Side SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

The side between two angles

GI

GH

HI


E SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Y

S

Included Side

Name the included side:

Y and E

E and S

S and Y

YE

ES

SY


Side side side congruence postulate
Side-Side-Side Congruence Postulate SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

SSS Post. - If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

If

then,


Using sss congruence post

1) Given SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

2) SSS

Using SSS Congruence Post.

Prove:

  • 1)

  • 2)


Side angle side congruence postulate
Side-Angle-Side Congruence Postulate SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

SAS Post. – If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

If

then,


Included Angle SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

The angle betweentwo sides

H

G

I


Included Angle SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Name the included angle:

YE and ES

ES and YS

YS and YE

E

E

S

Y

Y

S


Included Side SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

The sidebetween two angles

GI

GH

HI


E SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Y

S

Included Side

Name the included side:

Y and E

E and S

S and Y

YE

ES

SY


Triangle congruency short cuts1
Triangle congruency short-cuts SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write


Given: SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write HJ  GI, GJ  JI

Prove: ΔGHJ  ΔIHJ

HJ  GI Given

GJH & IJH are Rt <‘s

Def. ┴lines

GJH  IJH

Rt <‘s are ≅

GJ  JI Given

HJ  HJ Reflexive Prop

ΔGHJ  ΔIHJ SAS


Given: SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write 1  2, A  E and AC  EC

Prove: ΔABC  ΔEDC

1  2Given

A  E Given

AC  EC Given

ΔABC  ΔEDC ASA


Given: SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write ΔABD, ΔCBD, AB  CB,

and AD  CD

Prove: ΔABD  ΔCBD

AB  CBGiven

AD  CD Given

BD  BD Reflexive Prop

 ΔABD  ΔCBD SSS


Given: SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write LJ bisects IJK,

ILJ   JLK

Prove: ΔILJ  ΔKLJ

LJ bisects IJK Given

IJL  IJH Definition of bisector

ILJ   JLK Given

JL  JL Reflexive Prop

ΔILJ  ΔKLJ ASA


Given: SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write TV  VW, UV VX

Prove: ΔTUV  ΔWXV

TV  VWGiven

UV  VX Given

TVU  WVX Vertical angles

 ΔTUV  ΔWXV SAS


Given: SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write Given: HJ  JL, H L

Prove: ΔHIJ  ΔLKJ

HJ  JL Given

H L Given

IJH  KJL Vertical angles

 ΔHIJ  ΔLKJ ASA


Given: SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write Quadrilateral PRST with PR  ST,

PRT  STR

Prove: ΔPRT  ΔSTR

PR  STGiven

PRT  STR Given

RT  RT Reflexive Prop

ΔPRT  ΔSTR SAS


Given: SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write Quadrilateral PQRS, PQ  QR,

PS  SR, and QR  SR

Prove: ΔPQR  ΔPSR

PQ  QRGiven

PQR = 90° PQ  QR

PS  SRGiven

PSR = 90° PS  SR

QR  SR Given

PR  PR Reflexive Prop

ΔPQR  ΔPSR HL


Prove it

Prove it! SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

NOT triangle congruency short cuts


Not triangle congruency short cuts
NOT triangle congruency short-cuts SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • The following are NOT short cuts:

  • AAA (angle-angle-angle)

  • Triangles are similar but not necessarily congruent


Not triangle congruency short cuts1
NOT triangle congruency short-cuts SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • The following are NOT short cuts

  • SSA (side-side-angle)

  • SAS is a short cut but the angle is in between both sides!


Prove it1

Prove it! SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)


Cpctc
CPCTC SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • Once you have proved two triangles congruent using one of the short cuts, the rest of the parts of the triangle you haven’t proved directly are also congruent!

  • We say: Corresponding Parts of Congruent Triangles are Congruent or CPCTC for short


Cpctc example
CPCTC example SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Given: ΔTUV, ΔWXV, TV  WV,

TW bisects UX

Prove: TU  WX

Statements: Reasons:

  • TV  WV Given

  • UV  VX Definition of bisector

  • TVU  WVX Vertical angles are congruent

  • ΔTUV  ΔWXV SAS

  • TU  WX CPCTC


Side side side

AC PX SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

AB PN

CB XN

Therefore, using SSS,

∆ABC= ∆PNX

X

A

3 inches

3 inches

5 inches

7 inches

~

P

C

B

7 inches

N

5 inches

~

~

~

=

=

=

Side Side Side

If 2 triangles have 3 corresponding pairs of sides that are congruent, then the triangles are congruent.


Side angle side

X SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

CA XP

CB XN

C  X

Therefore, by SAS,

∆ABC ∆PNX

60°

A

3 inches

5 inches

3 inches

P

N

60°

C

B

5 inches

~

~

~

~

=

=

=

=

Side Angle Side

If two sides and the INCLUDED ANGLE in one triangle are congruent to two sides and INCLUDED ANGLE in another triangle, then the triangles are congruent.


Angle side angle

CA XP SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

A P

C X

Therefore, by ASA,

∆ABC ∆PNX

~

~

~

~

=

=

=

=

Angle Side Angle

If two angles and the INCLUDED SIDE of one triangle are congruent to two angles and the INCLUDED SIDE of another triangle, the two triangles are congruent.

X

60°

A

3 inches

3 inches

70°

70°

P

N

60°

C

B


Side angle angle

60 SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write °

70°

Side Angle Angle

Triangle congruence can be proved if two angles and a NON-included side of one triangle are congruent to the corresponding angles and NON-included side of another triangle, then the triangles are congruent.

60°

70°

5 m

5 m

These two triangles are congruent by SAA


Remembering our shortcuts
Remembering our shortcuts SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

SSS

ASA

SAS

SAA


Corresponding parts

B SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

F

That means that EG  CB

A

E

What is AC congruent to?

FE

G

C

Corresponding parts

When you use a shortcut (SSS, AAS, SAS, ASA, HL) to show that 2 triangles are ,

that means that ALL the corresponding parts are congruent.

EX: If a triangle is congruent by ASA (for instance), then all the other corresponding parts are .


Corresponding parts of congruent triangles are congruent
Corresponding parts of congruent triangles are congruent. SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Corresponding parts of congruent triangles are congruent.

Corresponding parts of congruent triangles are congruent.

Corresponding parts of congruent triangles are congruent.


C SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write orresponding Parts of Congruent Triangles are Congruent.

CPCTC

If you can prove congruence using a shortcut, then you KNOW that the remaining corresponding parts are congruent.

You can only use CPCTC in a proof AFTER you have proved congruence.


For example

Statements Reasons SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

AC  DF Given

C F Given

CB  FE Given

ΔABC ΔDEF SAS

AB  DE CPCTC

For example:

A

Prove: AB  DE

B

C

D

F

E


Using sas congruence
Using SAS Congruence SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Prove: Δ VWZ ≅ Δ XWY

SAS

PROOF

Given

Δ VWZ ≅ Δ XWY

Vertical Angles


Proof
Proof SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Given: MB is perpendicular bisector of AP

Prove:

  • 1) MB is perpendicular bisector of AP

  • 2) <ABM and <PBM are right <‘s

  • 3)

  • 4)

  • 5)

  • 6)

  • 1) Given

  • 2) Def of Perpendiculars

  • 3) Def of Bisector

  • 4) Def of Right <‘s

  • 5) Reflexive Property

  • 6) SAS


Proof1
Proof SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Given: O is the midpoint of MQ and NP

Prove:

  • 1) O is the midpoint of MQ and NP

  • 2)

  • 3)

  • 4)

  • 1) Given

  • 2) Def of midpoint

  • 3) Vertical Angles

  • 4) SAS


Proof2
Proof SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

  • 1)

  • 2)

  • 3)

  • 1) Given

  • 2) Reflexive Property

  • 3) SSS

Given:

Prove:


Proof3
Proof SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Given:

Prove:

  • 1)

  • 2)

  • 3)

  • 4)

  • 1) Given

  • 2) Alt. Int. <‘s Thm

  • 3) Reflexive Property

  • 4) SAS


Checkpoint
Checkpoint SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Decide if enough information is given to prove the triangles are congruent. If so, state the congruence postulate you would use.


Congruent triangles in the coordinate plane
Congruent Triangles in the Coordinate Plane SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Use the SSS Congruence Postulate to show that

∆ABC ≅ ∆DEF

Which other postulate could you use to prove the triangles are congruent?


Congruent Triangles SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write


2 SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

(

)

+

2

(

)

x

x

y

y

2

1

2

1

d

=

EXAMPLE 2

Standardized Test Practice

SOLUTION

By counting, PQ = 4 and QR = 3. Use the Distance Formula to find PR.


Write a proof. SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

GIVEN

KL NL,KM NM

PROVE

KLMNLM

Proof

KL NL andKM NM

It is given that

LM LM.

By the Reflexive Property,

So, by the SSS Congruence Postulate,

KLMNLM


DFG SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write HJK

SideDG HK, SideDF JH,andSideFG JK.

So by the SSS Congruence postulate, DFG HJK.

for Example 1

GUIDED PRACTICE

Decide whether the congruence statement is true. Explain your reasoning.

SOLUTION

Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent.

Yes. The statement is true.


Included Angle SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

The angle between two sides

H

G

I


E SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write

Y

S

Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

E

S

Y


In the diagram at the right, what postulate or theorem can you use to prove that

RSTVUT

Given

S U

Given

RS UV

Vertical angles

RTSUTV

Δ RST ≅ Δ VUT

SAA


Now For The Fun Part… you use to prove that

Proofs!


Given jo sh o is the midpoint of sh prove soj hoj
Given: JO you use to prove that SH; O is the midpoint of SH Prove:  SOJ  HOJ


A C you use to prove that

E

B D

Given: BC bisects AD

A D

Prove: AB  DC


WORK you use to prove that


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