G.6

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Proving Triangles Congruent

1

Two geometric figures with exactly the same size and shape.

The Idea of Congruence

A C

B

D E

F

2

How much do you need to know. . . . . . about two triangles to prove that they are congruent?

3

Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.

Corresponding Parts

ABC DEF

1. AB DE

2. BC EF

3. AC DF

4. A D

5. B E

6. C F

4

Do you need all six ?

NO !

SSS SAS ASA AAS HL

5

Side-Side-Side (SSS)

1. AB DE

2. BC EF

3. AC DF

ABC DEF

Side

Side

Side

The triangles are congruent by

SSS.

If the sides of one triangle are congruent to the sides of a

second triangle, then the triangles are congruent.

6

The angle between two sides

Included Angle

HGI G

GIH I

GHI H

This combo is called side-angle-side, or just SAS.

7

Name the included angle:

YE and ES

ES and YS

YS and YE

Included Angle

S Y

E

YES or E

YSE or S EYS or Y The other two

angles are the NON-INCLUDED

angles.

8

Side-Angle-Side (SAS)

1. AB DE

2. A D

3. AC DF

ABC DEF

included angle

Side

Angle

Side

The triangles are congruent by

SAS.

If two sides and the included angle of one triangle are

congruent to the two sides and the included angle of another

triangle, then the triangles are congruent.

9

The side between two angles

Included Side

GI HI GH

This combo is called angle-side-angle, or just ASA.

10

Name the included side:

Y and E

E and S

S and Y

Included Side

S Y

E

YE

ES

SY

The other two sides are the

NON-INCLUDED sides.

11

Angle-Side-Angle (ASA)

1. A D

2. AB DE

3. B E

ABC DEF

included side

Angle

Side

Angle

The triangles are congruent by

ASA.

If two angles and the included side of one triangle are

congruent to the two angles and the included side of another

triangle, then the triangles are congruent.

12

Angle-Angle-Side (AAS)

1. A D

2. B E

3. BC EF

ABC DEF

Non-included side Side

Angle

Angle

The triangles are congruent by

AAS.

If two angles and a non-included side of one triangle are

congruent to the corresponding angles and side of another

triangle, then the triangles are congruent.

13

Warning: No SSA Postulate

There is no such

thing as an SSA

postulate!

The triangles are NOTcongruent!

Side

Side

Angle

14

Warning: No SSA Postulate

NOT CONGRUENT!

There is no such

thing as an SSA

postulate!

15

BUT: SSA DOES work in one

situation!

If we know that

the two triangles

are right

triangles!

Side

Side

Side

Angle

16

We call this

These triangles ARE CONGRUENT by HL!

HL,

for “Hypotenuse – Leg”

Hypotenuse

Leg

Hypotenuse

RIGHT Triangles!

Remember! The

triangles must be RIGHT!

17

Hypotenuse-Leg (HL)

1.AB HL

2.CB GL

3.C and G are rt. ‘s

ABC DEF

The triangles are congruent

by HL.

Right Triangle

Leg

If the hypotenuse and a leg of a right triangle are congruent

to the hypotenuse and a leg of another right triangle, then the

triangles are congruent.

18

Warning: No AAA Postulate

A C

B

D

E

F

There is no such

thing as an AAA

postulate!

NOT CONGRUENT!

Same Shapes!

Different Sizes!

19

Congruence Postulates

and Theorems

• SSS • SAS • ASA • AAS • AAA? • SSA? • HL

20

Name That Postulate

SAS ASA

AAS SSA

(when possible)

Not enough info!

21

Name That Postulate (when possible)

SSS AAA

SSA

Not enough info!

Not enough info!

SSA HL

22

Name That Postulate (when possible)

SSA

AAA

Not enough info!

Not enough info!

HL

SSA

Not enough info!

23

Vertical Angles,

Reflexive Sides and Angles When two triangles touch, there may be additional congruent parts.

Vertical Angles

Reflexive Side

side shared by two

triangles

24

Name That Postulate (when possible)

SAS

AAS

SAS Reflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SSA

Not enough info!

25

When two triangles overlap, there may be additional congruent parts.

Reflexive Side side shared by two

triangles

Reflexive Angle angle shared by two

triangles

Reflexive Sides and Angles 26

Let’s Practice

Indicate 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

27

Try Some Proofs

End Slide Show

What’s Next

28

Choose a

Problem.

Problem #1

Problem #2

Problem #3

End Slide Show

D

A B

C

E

C

D

AB

Z

W Y

X

SSS

SAS

ASA

29

D

A B

C

Given: AB CD

BC DAProve: ABC CDA

Problem #1

30

Step 1: Mark the Given

D

A B

C

Given: AB CD

BC DAProve: ABC CDA

31

•Reflexive Sides

•Vertical Angles

D

A B

C

Given: AB CD

BC DAProve: ABC CDA

Step 2: Mark . . .

… if they exist.

32

Step 3: Choose a Method

SSS

SAS

ASA

AAS

HL

D

A B

C

Given: AB CD

BC DAProve: ABC CDA

33

Step 4: List the Parts

D

A B

C

Given: AB CD

BC DAProve: ABC CDA

STATEMENTS REASONS

1. AB CD

2. BC DA

3. AC AC

… in the order of the Method

S

S

S

34

Step 5: Fill in the Reasons

(Why did you mark those parts?)

D

A B

C

Given: AB CD

BC DAProve: ABC CDA

STATEMENTS REASONS

1. AB CD

2. BC DA

3. AC AC

1. Given2. Given3. Reflexive Prop.

S

S

S

35

S

S

S

Step 6: Is there more?

D

A B

C

Given: AB CD

BC DAProve: ABC CDA

STATEMENTS REASONS

1. AB CD

2. BC DA

3. AC AC

1. Given2. Given3. Reflexive Prop.

4. ABC CDA 4. SSS (pos.)

The “Prove” Statement

is always last !

36

Choose a

Problem.

Problem #1

Problem #2

Problem #3

End Slide Show

D

A B

C

E

C

D

AB

Z

W Y

X

SSS

SAS

ASA

37

Problem #2

E

C

D

AB

Given: AB CB

EB DBProve: ABE CBD

38

Step 1: Mark the Given

E

C

D

AB

Given: AB CB

EB DBProve: ABE CBD

39

•Reflexive Sides

•Vertical Angles

… if they exist.

Step 2: Mark . . .

E

C

D

AB

Given: AB CB

EB DBProve: ABE CBD

40

Step 3: Choose a Method

SSS

SAS

ASA

AAS

HL

E

C

D

AB

Given: AB CB

EB DBProve: ABE CBD

41

Step 4: List the Parts

… in the order of the Method

STATEMENTS REASONS

E

C

D

AB

Given: AB CB

EB DBProve: ABE CBD

1. AB CB2. ABE CBD

3. EB DB

S

A

S

42

Step 5: Fill in the Reasons

(Why did you mark those parts?)

STATEMENTS REASONS

E

C

D

AB

Given: AB CB

EB DBProve: ABE CBD

1. AB CB2. ABE CBD

3. EB DB

1. Given2. Vertical s (thm.)

3. Given

S

A

S

43

Step 6: Is there more?

STATEMENTS REASONS

E

C

D

AB

Given: AB CB

EB DBProve: ABE CBD

4. ABE CBD

1. AB CB2. ABE CBD

3. EB DB

1. Given2. Vertical s (thm.)

3. Given4. SAS (pos.)

The “Prove” Statement

is always last !

S

A

S

44

Choose a

Problem.

Problem #1

Problem #2

Problem #3

End Slide Show

D

A B

C

E

C

D

AB

Z

W Y

X

SSS

SAS

ASA

45

Problem #3

Z

W Y

XGiven: XWY ZWY XYW ZYWProve: WXY WZY

46

Step 1: Mark the Given

Z

W Y

XGiven: XWY ZWY XYW ZYWProve: WXY WZY

47

•Reflexive Sides

•Vertical Angles

… if they exist.

Step 2: Mark . . .

Z

W Y

XGiven: XWY ZWY XYW ZYWProve: WXY WZY

48

Step 3: Choose a Method

SSS

SAS

ASA

AAS

HL

Z

W Y

XGiven: XWY ZWY XYW ZYWProve: WXY WZY

49

Step 4: List the Parts

… in the order of the Method

STATEMENTS REASONS

Z

W Y

XGiven: XWY ZWY XYW ZYWProve: WXY WZY

1. XWY ZWY

2. WY WY3. XYW ZYW

A

S

A

50

Step 5: Fill in the Reasons

(Why did you mark those parts?)

STATEMENTS REASONS

Z

W Y

XGiven: XWY ZWY XYW ZYWProve: WXY WZY

1. XWY ZWY

2. WY WY3. XYW ZYW

1. Given2. Reflexive (pos.)

3. Given

A

S

A

51

Step 6: Is there more?

STATEMENTS REASONS

Z

W Y

XGiven: XWY ZWY XYW ZYWProve: WXY WZY

1. XWY ZWY

2. WY WY3. XYW ZYW

4. WXY WZY

1. Given2. Reflexive (pos.)

3. Given

4. ASA (pos.)

The “Prove” Statement

is always last !

A

S

A

52

Choose a

Problem.

Problem #1

Problem #2

Problem #3

End Slide Show

D

A B

C

E

C

D

AB

Z

W Y

X

SSS

SAS

ASA

53

Choose a

Problem.

Problem #4

Problem #5

End Slide Show

AAS

HL

E

C

D

AB

CB D

A

54

Problem #4

Statements Reasons

AAS

Given

Given

Vertical Angles Thm

AAS Postulate

Given: A C

BE BDProve: ABE CBD

E

C

D

AB

4. ABE CBD

55

Choose a

Problem.

Problem #4

Problem #5

End Slide Show

AAS

HL

E

C

D

AB

CB D

A

56

Problem #5

3. AC AC

Statements Reasons

CB D

AHL

Given

Given

Reflexive Property

HL Postulate 4. ABC ADC

1. ABC, ADC right s

AB AD

Given ABC, ADC right s,

Prove:

AB AD

ABC ADC

57

Congruence Proofs 1. Mark the Given.

2. Mark …

Reflexive Sides or Angles / Vertical Angles

Also: mark info implied by given info.

3. Choose a Method. (SSS , SAS, ASA)

4. List the Parts …

in the order of the method.

5. Fill in the Reasons …

why you marked the parts.

6. Is there more?

58

Given implies Congruent Parts

midpoint

parallel

segment bisector

angle bisector

perpendicular

segments

angles

segments

angles

angles

59

Example Problem

CB D

AGiven: AC bisects BAD

AB ADProve: ABC ADC

60

Step 1: Mark the Given

… and

what it

implies

CB D

AGiven: AC bisects BAD

AB ADProve: ABC ADC

61

•Reflexive Sides

•Vertical Angles Step 2: Mark . . .

… if they exist.

CB D

AGiven: AC bisects BAD

AB ADProve: ABC ADC

62

Step 3: Choose a Method

SSS

SAS

ASA

AAS

HL

CB D

AGiven: AC bisects BAD

AB ADProve: ABC ADC

63

Step 4: List the Parts

STATEMENTS REASONS

… in the order of the Method

CB D

AGiven: AC bisects BAD

AB ADProve: ABC ADC

BAC DAC

AB AD

AC AC

S

A

S

64

Step 5: Fill in the Reasons

(Why did you mark those parts?)

STATEMENTS REASONS

CB D

AGiven: AC bisects BAD

AB ADProve: ABC ADC

BAC DAC

AB AD

AC AC

Given

Def. of Bisector

Reflexive (prop.)

S

A

S

65

S

A

S

Step 6: Is there more?

STATEMENTS REASONS

CB D

AGiven: AC bisects BAD

AB ADProve: ABC ADC

BAC DAC

AB AD

AC AC

Given

AC bisects BAD Given

Def. of Bisector

Reflexive (prop.)

ABC ADC SAS (pos.)

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

66

Midpoint implies segments.

Back

CB D

A

STATEMENTS REASONS

1. C is the midpoint

of BD

S

Given: C is the midpoint of BD

AB AD Prove: ABC ADC

2. BC CD 2. Def. of Midpoint

1. Given

… AB AD3. 3. Given

67

Parallel implies angles.

Back

STATEMENTS REASONS D

A B

C

Given: AB DC

AD BC Prove: ABC CDA

2. BAC DCA 2. Alt. Int. s (thm.)1. AB DC

3. AD BC 3. Given

4. DAC BCA 4. Alt. Int. s (thm.)

A

A

1. Given

68

Seg. bisector implies segments. Back

STATEMENTS REASONS

CB

A

DE

2. EB BC

4. AB BD

1. AD bisects EC 1. Given 2. Def. of bisect

3. EC bisects AD 3. Given 4. Def. of bisect

S

S

Given: AD bisects EC

EC bisects ADProve: ABE DBC

…

69

Angle bisector implies angles.

Back

STATEMENTS REASONS

CB D

AGiven: AC bisects BAD

AB ADProve: ABC ADC

2. BAC DAC

1. AC bisects BAD 1. Given

A

…

2. Def. of bisect

70

implies right ( ) angles. Back

STATEMENTS REASONS

CB D

AGiven: AC BD

BC DCProve: ABC ADC

2. ACB and ACD are right s

3. ACB ACD

1. AC BD 1. Given

2. lines form 4 rt. s (thm)

3. All rt. s are (thm)

A

… S 2. BC CD 4. Given 4.

71

Congruent Triangles Proofs

1. Mark the Given and what it implies.

2. Mark … Reflexive Sides / Vertical Angles

3. Choose a Method. (SSS , SAS, ASA)

4. List the Parts …

in the order of the method.

5. Fill in the Reasons …

why you marked the parts.

6. Is there more?

72

Using CPCTC in Proofs

According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent.

This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles.

This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.

73

Corresponding Parts of

Congruent Triangles

For example, can you prove that sides AD and BC are congruent in the figure at right?

The sides will be congruent if triangle ADM is congruent

to triangle BCM. Angles A and B are congruent because they are marked.

Sides MA and MB are congruent because they are marked.

Angles 1 and 2 are congruent because they are vertical angles.

So triangle ADM is congruent to triangle BCM by ASA.

This means sides AD and BC are congruent by CPCTC.

74

Corresponding Parts of

Congruent Triangles

A two column proof that sides AD and BC

are congruent in the figure at right is shown

below:

Statement Reason

MA MB Given

A B Given

1 2 Vertical angles

ADM BCM ASA

AD BC CPCTC

75

Corresponding Parts of

Congruent Triangles

A two column proof that sides AD and BC

are congruent in the figure at right is shown

below:

Statement Reason

MA MB Given

A B Given

1 2 Vertical angles

ADM BCM ASA

AD BC CPCTC

76

Corresponding Parts of

Congruent Triangles

Sometimes it is necessary to add an auxiliary

line in order to complete a proof

For example, to prove ÐR @ ÐO in this picture

Statement Reason

FR @ FO Given

RU @ OU Given

UF @ UF reflexive prop.

DFRU @ DFOU SSS

ÐR @ ÐO CPCTC

77

Corresponding Parts of

Congruent Triangles

Sometimes it is necessary to add an auxiliary

line in order to complete a proof

For example, to prove ÐR @ ÐO in this picture

Statement Reason

FR @ FO Given

RU @ OU Given

UF @ UF Same segment

DFRU @ DFOU SSS

ÐR @ ÐO CPCTC

78