Lesson 4 4 4 5 proving triangles congruent
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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

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Lesson 4.4 - 4.5Proving 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!


Built – In Information in Triangles


Identify the ‘built-in’ part


Shared side

SSS

Vertical angles

SAS

Parallel lines

-> AIA

Shared side

SAS


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)


Closure Question


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

  • 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

Prove: ∆DEF ≅ ∆JKL

From the diagram,

SSS Congruence Postulate.

∆DEF ≅ ∆JKL


SAS (Side-Angle-Side) Congruence Postulate

  • 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

  • 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

Prove: ∆SYT ≅ ∆WYX


Side-Side-Side Postulate

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


Angle-Angle-Side Postulate

  • 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

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


Side-Angle-Side Postulate

  • 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?

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

  • 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

  • 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

  • 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

  • Is it possible to prove these triangles are congruent?


Example

  • Is it possible to prove these triangles are congruent?

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


Example

  • Is it possible to prove these triangles are congruent?


Example

  • Is it possible to prove these triangles are congruent?

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


Example

  • Is it possible to prove these triangles are congruent?

2

1

3

4


Example

  • 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

The angle between two sides

H

G

I


E

Y

S

Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

E

S

Y


Included Side

The side between two angles

GI

GH

HI


E

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

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,


1) Given

2) SSS

Using SSS Congruence Post.

Prove:

  • 1)

  • 2)


Side-Angle-Side Congruence Postulate

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

The angle betweentwo sides

H

G

I


Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

E

E

S

Y

Y

S


Included Side

The sidebetween two angles

GI

GH

HI


E

Y

S

Included Side

Name the included side:

Y and E

E and S

S and Y

YE

ES

SY


Triangle congruency short-cuts


Given: HJ  GI, GJ  JI

Prove: ΔGHJ  ΔIHJ

HJ  GIGiven

GJH & IJH are Rt <‘s

Def. ┴lines

GJH  IJH

Rt <‘s are ≅

GJ  JIGiven

HJ  HJReflexive Prop

ΔGHJ  ΔIHJSAS


Given: 1  2, A  E and AC  EC

Prove: ΔABC  ΔEDC

1  2Given

A  E Given

AC  EC Given

ΔABC  ΔEDCASA


Given: ΔABD, ΔCBD, AB  CB,

and AD  CD

Prove: ΔABD  ΔCBD

AB  CBGiven

AD  CD Given

BD  BDReflexive Prop

 ΔABD  ΔCBDSSS


Given: LJ bisects IJK,

ILJ   JLK

Prove: ΔILJ  ΔKLJ

LJ bisects IJKGiven

IJL  IJHDefinition of bisector

ILJ   JLK Given

JL  JLReflexive Prop

ΔILJ  ΔKLJASA


Given:TV  VW, UV VX

Prove: ΔTUV  ΔWXV

TV  VWGiven

UV  VX Given

TVU  WVXVertical angles

 ΔTUV  ΔWXV SAS


Given: Given: HJ  JL, H L

Prove: ΔHIJ  ΔLKJ

HJ  JLGiven

H L Given

IJH  KJLVertical angles

 ΔHIJ  ΔLKJ ASA


Given: Quadrilateral PRST with PR  ST,

PRT  STR

Prove: ΔPRT  ΔSTR

PR  STGiven

PRT  STR Given

RT  RTReflexive Prop

ΔPRT  ΔSTR SAS


Given: 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  SRGiven

PR  PRReflexive Prop

ΔPQR  ΔPSRHL


Prove it!

NOT triangle congruency short cuts


NOT triangle congruency short-cuts

  • The following are NOT short cuts:

  • AAA (angle-angle-angle)

  • Triangles are similar but not necessarily congruent


NOT triangle congruency short-cuts

  • The following are NOT short cuts

  • SSA (side-side-angle)

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


Prove it!

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)


CPCTC

  • 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

Given: ΔTUV, ΔWXV, TV  WV,

TW bisects UX

Prove: TU  WX

Statements:Reasons:

  • TV  WVGiven

  • UV  VXDefinition of bisector

  • TVU  WVXVertical angles are congruent

  • ΔTUV  ΔWXVSAS

  • TU  WXCPCTC


AC PX

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.


X

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.


CA XP

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


60°

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

SSS

ASA

SAS

SAA


B

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.

Corresponding parts of congruent triangles are congruent.

Corresponding parts of congruent triangles are congruent.


Corresponding 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.


Statements Reasons

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

Prove: Δ VWZ ≅ Δ XWY

SAS

PROOF

Given

Δ VWZ ≅ Δ XWY

Vertical Angles


Proof

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


Proof

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


Proof

  • 1)

  • 2)

  • 3)

  • 1) Given

  • 2) Reflexive Property

  • 3) SSS

Given:

Prove:


Proof

Given:

Prove:

  • 1)

  • 2)

  • 3)

  • 4)

  • 1) Given

  • 2) Alt. Int. <‘s Thm

  • 3) Reflexive Property

  • 4) SAS


Checkpoint

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

Use the SSS Congruence Postulate to show that

∆ABC ≅ ∆DEF

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


Congruent Triangles


2

(

)

+

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.

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


DFGHJK

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

The angle between two sides

H

G

I


E

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…

Proofs!


Given: JO  SH; O is the midpoint of SH Prove:  SOJ  HOJ


A C

E

B D

Given: BC bisects AD

A D

Prove: AB  DC


WORK


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