lesson 4.4 - 4.5 proving triangles congruent triangle congruency short-cuts if you can prove one of...

Lesson 4.4 - 4.5Proving Triangles

Congruent

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

1.SSS (side-side-side)2.SAS (side-angle-side)3.ASA (angle-side-angle)4.AAS (angle-angle-side)5.HL (hypotenuse-leg) right triangles only!

Built – In Information in Triangles

Identify the ‘built-in’ part Identify the ‘built-in’ part

Shared sideShared side

Parallel lines Parallel lines -> AIA-> AIA

Shared sideShared side

Vertical anglesVertical angles

SASSAS

SASSAS

SSSSSS

SOME REASONS For Indirect SOME REASONS For Indirect InformationInformation

• Def of midpointDef of midpoint• Def of a bisectorDef of a bisector• Vert angles are congruentVert angles are congruent• Def of perpendicular bisectorDef of perpendicular bisector• Reflexive property (shared side)Reflexive property (shared side)• Parallel lines ….. alt int anglesParallel lines ….. alt int angles• Property of Perpendicular LinesProperty of Perpendicular Lines

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

1. AB DE

2. BC EF

3. AC DF

ABC DEF

B

A

C

E

D

F

Side-Angle-Side (SAS)Side-Angle-Side (SAS)

1. AB DE

2. A D

3. AC DF

ABC DEF

B

A

C

E

D

F

included angle

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

1. A D

2. AB DE

3. B E

ABC DEF

B

A

C

E

D

F

included side

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

1. A D

2. B E

3. BC EF

ABC DEF

B

A

C

E

D

F

Non-included

side

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!

Name That PostulateName That Postulate

SASSASASAASA

SSSSSSSSASSA

(when possible)

This is called a common side.This is called a common side.It is a side for both triangles.It is a side for both triangles.

We’ll use the reflexive property.We’ll use the reflexive property.

HLHL ( hypotenuse leg ) is used( hypotenuse leg ) is usedonly with right triangles, BUT, only with right triangles, BUT,

not all right triangles. not all right triangles.

HLHL ASAASA

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

Closure Question

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

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.

ΔGIH ΔJIK by AAS

G

I

H J

KEx 4

ΔABC ΔEDC by ASA

B A

C

ED

Ex 5


ΔACB ΔECD by SAS

B

A

C

E

D

Ex 6


ΔJMK ΔLKM by SAS or ASA

J K

LM

Ex 7


Not possible

K

J

L

T

U

Ex 8


V

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 PQAB QRBC PRAC

∆ ABC ≅ ∆PQR

Then

Side

Side

Example 1

Prove: ∆DEF ≅ ∆JKL

From the diagram,

.,, KLEFandJLDFJKDE

SSS Congruence Postulate.∆ DEF ≅ ∆JKL

• 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.

SAS (Side-Angle-Side) Congruence Postulate

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. If Angle

DFAC

∆ABC ∆≅ DEF

Then

Side

Angle

∠A D≅ ∠

∠C F≅ ∠

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

A

B

• 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.

C

Q

RS

AAS

A

B

• 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.

C

Q

RS

AAS Proof

A

B

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

• Third Angle Theorem

• Now QR is an included side, so ASA.

C

Q

RS

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?

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



12

34

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.

12

34

The angle between two sides

Included AngleIncluded Angle

G I H

Name the included angle:

YE and ES

ES and YS

YS and YE


SY

E

E

S

Y

The side between two angles

Included SideIncluded Side

GI HI GH

Name the included side:

Y and E

E and S

S and Y


SY

E

YE

ES

SY

Side-Side-Side Congruence Postulate

, ,MN QR NP RS PM SQ MNP QRS

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.PQW TSW

• 1)• 2)

• 1) Given• 2) SSS

Prove:

, ,PQ ST QW WS PW TW PQW TSW

Side-Angle-Side Congruence Postulate

, ,PQ WX QS XY Q X PQS WXY

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,



G I H


YE and ES

ES and YS

YS and YE


SY

E

E

S

Y

The side between two angles


GI HI GH

Name the included side:

Y and E

E and S

S and Y


SY

E

YE

ES

SY

Triangle congruency short-cuts

A B

C

D

EF

Given: HJ GI, GJ JIProve: ΔGHJ ΔIHJ

HJ GI GivenGJH & IJH are Rt <‘s

Def. ┴ lines

GJH IJHRt <‘s are ≅

GJ JI GivenHJ HJ Reflexive Prop

ΔGHJ ΔIHJ SAS

JG

H

I

Given: 1 2, A E and AC EC Prove: ΔABC ΔEDC

1 2 GivenA E GivenAC EC Given

ΔABC ΔEDCASA

21

C

D

EA

B

Given: ΔABD, ΔCBD, AB CB, and AD CD

Prove: ΔABD ΔCBD

AB CB GivenAD CD Given BD BD Reflexive Prop

ΔABD ΔCBD SSS

B

C

A

D

Given: LJ bisects IJK, ILJ JLK

Prove: ΔILJ ΔKLJ

LJ bisects IJKGivenIJL IJH Definition of bisectorILJ JLK GivenJL JL Reflexive Prop

ΔILJ ΔKLJ ASA

J

K

I

L

Given: TV VW, UV VX Prove: ΔTUV ΔWXV

TV VW GivenUV VX GivenTVU WVX Vertical angles

ΔTUV ΔWXV SAS

VT

W

U

X

Given: Given: HJ JL, H L Prove: ΔHIJ ΔLKJ

HJ JL GivenH L GivenIJH KJL Vertical angles

ΔHIJ ΔLKJ ASA

L

J

KI

H

Given: Quadrilateral PRST with PR ST, PRT STR

Prove: ΔPRT ΔSTR

PR ST GivenPRT STR Given RT RT Reflexive Prop

ΔPRT ΔSTR SAS

S

P T

R

Given: Quadrilateral PQRS, PQ QR, PS SR, and QR SR

Prove: ΔPQR ΔPSR

PQ QR GivenPQR = 90° PQ QR PS SR GivenPSR = 90° PS SR QR SR GivenPR PR Reflexive Prop

ΔPQR ΔPSR HL

S

RP

Q

Prove it!Prove it!Prove it!Prove it!

NOT triangle congruency short NOT triangle congruency short cutscuts

NOT triangle congruency short-cuts

• The following are NOT short cuts:

• AAA (angle-angle-angle)

• Triangles are similar but not necessarily congruent

60

60

60

A

BC

60

60

60

D

F E

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!

5 cm8 cm

34

A

B

C

5 cm8 cm

34

D

E

F

Prove it!Prove it!Prove it!Prove it!

CPCTC (Corresponding Parts of CPCTC (Corresponding Parts of Congruent Triangles are Congruent Triangles are

Congruent)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 WXStatements: Reasons:1. TV WV Given2. UV VX Definition of bisector3. TVU WVX Vertical angles are congruent4. ΔTUV ΔWXV SAS5. TU WX CPCTC

VT

W

U

X

Side Side Side

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

A

B

C

3 in

ches 7 inches

5 inches

X

P

N

3 inches

7 inches

5 in

ches

AC PX

AB PN

CB XN

Therefore, using SSS,

∆ABC = ∆PNX

~

=~=~=~

Side Angle SideIf 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.

A

B

C

3 in

ches

5 inches

60°

X

P

N

3 inches5 in

ches

60°CA XP

CB XN

C X

Therefore, by SAS,

∆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.

A

B

C

3 in

ches

60°

70°

X

P

N

3 inches

60°

70°

CA XP

A P

C X

Therefore, by ASA,

∆ABC ∆PNX

=~=~

=~

=~

Side Angle AngleTriangle 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

60°

70°

5 m

These two triangles are congruent by SAA

Remembering our shortcuts

SSSASASASSAA

Corresponding partsWhen 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 .

A

C

B

GE

FThat means that EG CB

What is AC congruent to?

FE

Corresponding parts of congruent triangles are

congruent.Corresponding parts of congruent triangles are



congruent.

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

Corresponding Parts of Congruent Triangles are

Congruent.

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

CPCTC

For example:

Prove: AB DEA

F E

D

C B

Statements Reasons

AC DF Given

C F Given

CB FE Given

ΔABC ΔDEF SAS

AB DE CPCTC

Using SAS Congruence

SAS

Prove: Δ VWZ ≅ Δ XWY,WZ WY VW XW

VWZ XWY Given

Vertical AnglesΔ VWZ ≅ Δ XWY

PROOF

Proof

• 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

Given: MB is perpendicular bisector of AP

Prove: ABM PBM

ABM PBM

AB BPABM PBM

BM BM

Proof

• 1) O is the midpoint of MQ and NP

• 2)

• 3)

• 4)

• 1) Given

• 2) Def of midpoint

• 3) Vertical Angles

• 4) SAS

Given: O is the midpoint of MQ and NP

Prove: MON POQ

MON POQ

,MO OQ NO OP MON POQ

Proof

• 1)

• 2)

• 3)

• 1) Given

• 2) Reflexive Property

• 3) SSS

Given:

Prove: ABC ADC

ABC ADC AC AC

,AB CD BC AD

,AB CD BC AD

Proof

• 1)

• 2)

• 3)

• 4)

• 1) Given

• 2) Alt. Int. <‘s Thm

• 3) Reflexive Property

• 4) SAS

ABD CDB

ABD CDB

, ||AD CB AD CB

, ||AD CB AD CB

DB DBADB CBD

Given:

Prove:

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?

EXAMPLE 2 Standardized Test Practice

SOLUTION

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

d = y2 – y1( )2x2 – x1( )2 +

Write a proof.

GIVEN KL NL, KM NM

Proof It is given that KL NL and KM NM

By the Reflexive Property, LM LM.

So, by the SSS Congruence Postulate, KLM NLM

PROVE KLM NLM

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

GUIDED PRACTICE for Example 1

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

SOLUTION

Yes. The statement is true.

1. DFG HJK

Side DG HK, Side DF JH,and Side FG JK.

So by the SSS Congruence postulate, DFG HJK.



G I H


YE and ES

ES and YS

YS and YE


SY

E

E

S

Y

GivenS U

Vertical anglesRTS UTV

GivenRS UV

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

RST VUT

Δ RST ≅ Δ VUT SAA

Now For The Fun Part…

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

J

S H0

Given: BC bisects AD A D

Prove: AB DC

A C

E

B D

lesson 4.4 - 4.5 proving triangles congruent triangle congruency short-cuts if you can prove one of...

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pair of triangles

congruent trianglessss

angleaas angleangle

anglesideasa angle

sss congruence

sas ab

b dfor aas

def jkl