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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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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!
slide5

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
slide7

Side-Side-Side (SSS)

E

B

F

A

D

C

  • AB DE
  • BC EF
  • AC DF

ABC DEF

slide8

Side-Angle-Side (SAS)

B

E

F

A

C

D

  • AB DE
  • A D
  • AC DF

ABC DEF

included

angle

slide9

Angle-Side-Angle (ASA)

B

E

F

A

C

D

  • A D
  • AB  DE
  • B E

ABC DEF

included

side

slide10

Angle-Angle-Side (AAS)

B

E

F

A

C

D

  • A D
  • B E
  • BC  EF

ABC DEF

Non-included

side

slide11

Warning: No AAA Postulate

There is no such thing as an AAA postulate!

E

B

A

C

F

D

NOT CONGRUENT

slide12

Warning: No SSA Postulate

There is no such thing as an SSA postulate!

E

B

F

A

C

D

NOT CONGRUENT

slide13

Name That Postulate

(when possible)

SAS

ASA

SSA

SSS

slide14

This is called a common side.

It is a side for both triangles.

We’ll use the reflexive property.

slide15

HL( hypotenuse leg ) is used

only with right triangles, BUT,

not all right triangles.

ASA

HL

slide16

Name That Postulate

(when possible)

Vertical Angles

Reflexive Property

SAS

SAS

Reflexive Property

Vertical Angles

SSA

SAS

slide17

Name That Postulate

(when possible)

slide18

Name That Postulate

(when possible)

slide20

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:

slide21

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

slide22

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

slide23

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

slide24

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

slide25

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
  • 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

Prove: ∆DEF ≅ ∆JKL

From the diagram,

SSS Congruence Postulate.

∆DEF ≅ ∆JKL

sas side angle side congruence postulate
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.
slide29

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
Example 2

Prove: ∆SYT ≅ ∆WYX

side side side postulate
Side-Side-Side Postulate
  • SSS postulate: If two triangles have three congruent sides, the triangles are congruent.
angle angle side postulate
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
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
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
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.

slide36
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

slide37
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
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
Example
  • Is it possible to prove these triangles are congruent?
example1
Example
  • Is it possible to prove these triangles are congruent?
  • Yes - vertical angles are congruent, so you have ASA
example2
Example
  • Is it possible to prove these triangles are congruent?
example3
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
example4
Example
  • Is it possible to prove these triangles are congruent?

2

1

3

4

example5
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

slide45

Included Angle

The angle between two sides

H

G

I

slide46

E

Y

S

Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

E

S

Y

slide47

Included Side

The side between two angles

GI

GH

HI

slide48

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
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,

side angle side congruence postulate
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,

slide52

Included Angle

The angle betweentwo sides

H

G

I

slide53

Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

E

E

S

Y

Y

S

slide54

Included Side

The sidebetween two angles

GI

GH

HI

slide55

E

Y

S

Included Side

Name the included side:

Y and E

E and S

S and Y

YE

ES

SY

slide57
Given: 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

slide58
Given: 1  2, A  E and AC  EC

Prove: ΔABC  ΔEDC

1  2Given

A  E Given

AC  EC Given

ΔABC  ΔEDC ASA

slide59
Given: ΔABD, ΔCBD, AB  CB,

and AD  CD

Prove: ΔABD  ΔCBD

AB  CBGiven

AD  CD Given

BD  BD Reflexive Prop

 ΔABD  ΔCBD SSS

slide60
Given: 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

slide61
Given:TV  VW, UV VX

Prove: ΔTUV  ΔWXV

TV  VWGiven

UV  VX Given

TVU  WVX Vertical angles

 ΔTUV  ΔWXV SAS

slide62
Given: Given: HJ  JL, H L

Prove: ΔHIJ  ΔLKJ

HJ  JL Given

H L Given

IJH  KJL Vertical angles

 ΔHIJ  ΔLKJ ASA

slide63
Given: Quadrilateral PRST with PR  ST,

PRT  STR

Prove: ΔPRT  ΔSTR

PR  STGiven

PRT  STR Given

RT  RT Reflexive Prop

ΔPRT  ΔSTR SAS

slide64
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  SR Given

PR  PR Reflexive Prop

ΔPQR  ΔPSR HL

prove it

Prove it!

NOT triangle congruency short cuts

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 cuts1
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 it1

Prove it!

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

cpctc
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
CPCTC example

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

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

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

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°

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

corresponding parts

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.

slide78

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.

for example

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
Using SAS Congruence

Prove: Δ VWZ ≅ Δ XWY

SAS

PROOF

Given

Δ VWZ ≅ Δ XWY

Vertical Angles

proof
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
proof1
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
proof2
Proof
  • 1)
  • 2)
  • 3)
  • 1) Given
  • 2) Reflexive Property
  • 3) SSS

Given:

Prove:

proof3
Proof

Given:

Prove:

  • 1)
  • 2)
  • 3)
  • 4)
  • 1) Given
  • 2) Alt. Int. <‘s Thm
  • 3) Reflexive Property
  • 4) SAS
checkpoint
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
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?

slide89

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.

slide90

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

slide91

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.

slide92

Included Angle

The angle between two sides

H

G

I

slide93

E

Y

S

Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

E

S

Y

slide94

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

slide97

A C

E

B D

Given: BC bisects AD

A D

Prove: AB  DC