Similarity and Congruence - intranet.cesc.vic.edu.au · Similarity and Congruence Curriculum Ready Similarity and Congruence ACMMG: 201, 220, 221, 243, 244

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Similarity and Congruence

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Similarity andCongruence

ACMMG: 201, 220, 221, 243, 244

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J 1100% Similarity and Congruence

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10

SIMILARITY AND CONGRUENCE

Try to answer these questions now, before working through the chapter.

Answer these questions, after working through the chapter.

If two shapes are congruent, it means thay are equal in every way – all their corresponding sides and angles are equal. Similar figures have the same shape, but not necessarily the same size. In this book, it is shown how similar and congruent shapes can be useful in solving problems.

But now I think:

What do I know now that I didn’t know before?

I used to think:

The symbol for congruent is /. What do you think it means to say TABC / TDEF?

The symbol for congruent is /. What do you think it means to say TABC / TDEF?

If two triangles are the same except for one angle, are they congruent?

If two triangles are the same except for one angle, are they congruent?

If a square with side length 4cm has been enlarged by a scale factor of 2, then what is the side length of the large square?

If a square with side length 4cm has been enlarged by a scale factor of 2, then what is the side length of the large square?

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J 102 100% Similarity and Congruence


Similarity and Congruence Basics

Congruent Triangles (/)

Congruent triangles are shapes that are exactly the same in every way (side lengths and interior angles are all equal). If even one side or one angle are not equal, then the triangles are not congruent.

Congruent Triangles These triangles are not congruent

All sides and angles are equal therefore triangles are congruent.

In ∆ABC and ∆MNP:

angles

From above, ∆ABC and ∆PQR are congruent. Using the proper notation, this is written as ∆ABC / ∆PQR. It is important to make sure the angles match when using the / symbol. Here is an example:

Notice the order of the letters when using /. The equal angles are written in the same order.

Equal angles written in the same order (correct):

Equal angles not written in the same order (incorrect):

No angle in DEFT is equal to N+ .` These triangles are NOT congruent.

Show that these triangles are congruent

B

11.4

7.4

10A

C

80c

60c

40c R

11.4

7.410

PQ

80c

60c40c

Angles: A P

B Q

C R

+ +

+ +

+ +

=

=

=

Sides: AB PQ

BC QR

CA RP

=

=

=

24.5

22.6

20

E

D

F

50c

70c

60c22.6

M

N P48c

Angle is different

10 14

15B C

A

75c

65c 40c

10 14

15N P

M

75c

65c 40c

A M

B N

C P

75

65

40

+ +

+ +

+ +

= =

= =

= =

c

c

c

sides

AB MN

BC NP

AC MP

=

=

=

ABC MNP

BCA NPM

CAB PMN

T T

T T

T T

/

/

/

ABC PMN

CAB PNM

BCA MNP

T T

T T

T T

/

/

/

` ∆ABC / ∆MNP

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10

Basics

Similar Figures |||( )

Similar figures have the same shape, but not necessarily the same size. These shapes are similar.

Find the length of AD.

Find the size of R+ . Find the size of RS.

Find the size of B+ .

PQRS and ABCD are similar.

PQRS and ABCD are similar. PQRS and ABCD are similar.

PQRS and ABCD are similar.

These shapes are similar

Similar figures have two important properties:

• Their corresponding angles are equal.

• Their corresponding sides are in the same ratio. In the above similar shapes, the ratio of the corresponding sides is 2 since the sides in the bigger shape are double the length in the smaller shape.

a

c d

b

D

C

10 cm

15 cm

5 cm 9 cm

K

L

M

J

10 cm

20 cm

18 cm

30 cm

A

B

P

S R

Q

3 cm

8 cm

45c

A

D C

B24 cm

18 cm135c

cm

PSAD

PQAB

AD

AD

3 824

9

`

`

`

=

=

=

B Q

B 45

`

`

+ +

+

=

= c

R C

R 135

`

`

+ +

+

=

= c

cm

DCRS

ABPQ

RS

RS

18 248

6

`

`

`

=

=

=

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Similarity and Congruence BasicsQuestions

1. Show these triangles are congruent, and then use / symbol to state congruency.

10

8

14

V U

T

10

8

14

E

F

G

13

10

13

A

B C67c

10

13

D

E F67c 67c

A

C B60c 25c

D

F E

95c

25c

a

c

b

d

12 13

5

P

QR

23c

67c

5

12

13

M

L

K

23c

67c

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10

Questions Basics

2. Find the missing values in these similar shapes (all measurements in cm):

Given EFLM 2= . Find the length of KN.

a

PQ = QR =

RS = P+ =

S+ = Q+ =

PT =

D+ =

QR =

ST =

P+ =

C+ =

LM = AB =

M+ = A+ =

c d

b

S

P

Q

R

14

D

E

F

G

76

9

10

110c

70c

95c

85c T

P

Q

R

S100c

50c

E

A

B

C

D

6

9 3

15

18

12

110c

130c

150c

L

K

M

J4

8

40cA

B

C

D

12

15

45c

E F

GH5

K

L

M

N

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Similarity and Congruence Knowing More

Testing for Congruent Triangles

Side Side Side (SSS)

If the corresponding sides of two triangles are equal, then the triangles are

congruent (SSS).

Side Angle Angle (SAA)

If 2 corresponding sides and a corresponding angle are equal, then the triangles are

congruent (SAA).

Side Angle Side (SAS)

If 2 sides and the included angle are respectively equal, then the triangles are

congruent (SAS).

Right Angle, Hypotenuse, Side (RHS)

If two right angled triangles have the same hypotenuse, and a corresponding side, then the

triangles are congruent (RHS).

In ∆ABC and ∆PQR:

In ∆ABC and ∆KLM:

In ∆ABC and ∆LMN:

In ∆ABC and ∆NOM:

Congruent triangles have all 3 corresponding sides equal, and all 3 corresponding angles equal – that is 6 properties. However, there are tests for congruent triangles that don’t require showing all 6 properties. There are four tests:

AB PQ

BC QR

AC PR

=

=

=

AB KL

A K

C M

+ +

+ +

=

=

=

AC NM

AB NO

C M 90c+ +

=

=

= =

AB LM

A L

AC LN

+ +

=

=

=

ABC PQRT T/

ABC KLMT T/ ABC NOMT T/

ABC LMNT T/(SSS)

(SAA) (RHS)

(SAS)

B

CA

Q

P R

B

CA

M

L N

A

B

C

K

L

M

A

BC

N

OM

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10

Knowing More

Here are some examples:

Here is an example where congruence is used to show something is true.

Show that these triangles are congruent:

Show that BD bisects +ABC in the diagram below

a

b

L

N

M93c

12 cm 12 cm25c

62c

I

K

J

93c

In IJKT

In LMNT and IJKT :

In DEFT and GEFT :

In ABDT and CBDT :

180 93 62J

25

c c c

c

+ = - -

=

N J` + +=

cm

93

12

N J

M K

MN KJ

LMN IKJ`

c

+ +

+ +

T T/

=

= =

= =

is common

DF GF

DE GE

EF

DEF GEF` T T/

=

=

(Angle sum of triangle)

(Proved above)

(Given)

(Given)

(Given)

(SSS)

(Given)

(Given)

(Congruent triangles, ABD CBDT T/ )

(RHS)

(Given)

(SAA)

(Both are 25c )

D

E

F

G

B

DA C

is common

bisecting

90

AB BC

BD

ADB CDB

ABD CBD

ABD CBD

BD ABC

`

`

`

c+ +

+ +

+ +

+

/

=

= =

=

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Similarity and Congruence Knowing More

There are two ways to show that triangles are similar:

• Show that their corresponding sides are in proportion. • Show that they have equal angles (AAA).

If two triangles are similar, the symbol ||| is used.

Similar Triangles |||( )

Show that these triangles are similar:

A

B C

H

G

I

58c

44c

Q

S

R

12

22

18

T

U

V

18

33

27

In ∆QRS and ∆TUV:

` All sides in proportion

` ∆QRS ||| ∆TUV

a

b

78c

58c

In ∆ABC:

+C = 180c - 58c - 78c = 44c

In ∆GHI:

+G = 180c - 58c - 44c = 78c

In ∆ABC and ∆GHI:

+C = +H

+A = +G

+B = + I

` ∆ABC ||| ∆GIH

(Angle sum of a triangle)

(Angle sum of a triangle)

(AAA)

(Both are 58c )

(Both are 78c )

(Both are 44c )

RQTU

RSUV

QSTV

1218

23

2233

23

1827

23

= =

= =

= =

RQTU

RSUV

QSTV= =c m

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10

Questions Knowing More

/

1. Explain what the following mean:

SSS

SAS

SAA

RHS

|||

a

b

c

d

e

f

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Similarity and Congruence Knowing MoreQuestions

c

2. Prove these triangles are congruent:

b

aC

AB

5

4

E

F D

5

3

A

B

C

D

E

M N

P

75c

S R

Q

75c 75c

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10

Questions Knowing More

3. In the diagram below, show that ∆BCD ||| ∆ACE.

4. Prove that ∆JKL ||| ∆STU.

A E

B D

C

J

KL

8

6

S

T

U

15

12

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Similarity and Congruence Using Our Knowledge

Scale Factor in Similar Triangles

When triangles are similar, their angles are equal (AAA) and their corresponding sides are in proportion. The ratio that their sides are in proportion is called the Scale Factor. A Scale Factor either enlarges (scales up) or reduces (scales down).

In ∆ABC and ∆DEF: In ∆ABC and ∆GHI:

If the scale factor is bigger than 1, the triangle is enlarged. If the scale factor is between 0 and 1 (decimal or fraction), the triangle is reduced.

In ∆LMN and ∆PQR:

` ∆LMN ||| ∆PQR

` The scale factor of ∆PQR to ∆LMN is 3.

(Corresponding sides are in proportion)

Show these triangles are similar and find their scale factor of ∆PQR to ∆LMN

Q

R

P

9cm

18cm

21cm

N

L

M

3cm

6cm

7cm

B

A

C

46

8

D

EF

3

4

2

H

G

I

812

16

Scale factor =2Scale factor 21=

ACDF

BCEF

ABDE

63

21

84

21

42

21

= =

= =

= =

| | |ABC DEF` T T | | |ABC GHI` T T

scale factor of ∆DEF to ∆ABC

scale factor of ∆GHI to ∆ABC

ACGI

BCHI

ABGH

612 2

816 2

48 2

= =

= =

= =

LMPQ

MNQR

NLRP

721 3

39 3

618 3

= =

= =

= =

3LMPQ

MNQR

NLRP= = =

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10

Using Our Knowledge

Using Similar Triangles

If triangles are known to be similar, then the properties of similar triangles can be used to solve problems.

Find the values of x and y if ∆ABC ||| ∆TUV (all measurements in cm)

T

U

V

y

18

30

A

x

B

C

10

5

To find x:

To find y:

In ∆ABC and ∆TUV:

(Similar triangles, ∆ABC ||| ∆TUV)

(Similar triangles, ∆ABC ||| ∆TUV)

cm

TVAC

TUAB

x

x

x

18 3010

3018 10

6

`

` #

`

=

=

=

=

cm

10

BCUV

ABTU

y

y

y

530

105 30

15

`

` #

`

=

=

=

=

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Similarity and Congruence Using Our KnowledgeQuestions

1. Find the scale factor in these pairs of similar triangles for both the smaller and larger triangles:

Given ∆ABE ||| ∆ACD.b

a

R

S

T

50

35

25

Given ∆RST ||| ∆UVW.

UV

W

10

7

5

A

B

C

DE

8

4

63

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10

Questions Using Our Knowledge

2. Answer these questions about the diagram below:

a

b

c

Show that ∆JML ||| ∆JNK.

Find the length of KL.

Find the length of ML.

J

K

LM

N

10 12

85

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Similarity and Congruence Using Our KnowledgeQuestions

3. Answer these questions about the shape below:

a

b

c

d

Show that ∆GFH ||| ∆GIJ.

Find the length of GI.

Find the length of IJ.

Find the scale factor of the larger triangle with respect to the small triangle.

H

F

G

J

I

8 6

10

25

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10

Thinking More

Show that the diagonals of a parallelogram bisect each other

In the diagram below, prove that BE || CD if ACAB

ADAE=

Using Congruence and Similarity in Proofs

Congruence and similarity are used to prove properties of triangles, quadrilaterals and other shapes.

Similarity can also be used in proofs.

A

O

B

CD

Draw in diagonals (AC and BD) in the parallelogram ABCD:

Given:

Given:

| |

| |

AB CD

AB CD

AD BC

AD BC

=

=

| |BE CD

| |AB CD

To prove:

To prove:

AO = OC and BO = OD

Proof:

Proof:

In ∆AOB and ∆COD

and

CAB DCA

ABD BDC

` + +

+ +

=

=

and

AOB COD

AO OC BO OD

`

`

T T/

= =

AB = CD

` The diagonals of a parallelogram bisect each other.

(Given)

(Alternate angles; AB || CD)

(Alternate angles; AB || CD)

(Congruent triangles; AOB CODT T/ )

(SAA)

(Given)

A

D

E

C

B

ACAB

ADAE=

In ∆ABE and ∆ACD

ACAB

ADAE

ABE ACD

ACD ABE

`

`

T T

T

/

/

=

||BE CD`

(Given)

(Corresponding sides in proportion)

(Similar triangles; | | |ABE ACDT T )

(Given)

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Similarity and Congruence Thinking MoreQuestions

1. Answer these questions about PQRS below given that PQ = RS and PR = QS:

a

b

c

Prove ∆PRS / ∆SQP.

Prove PQRS is a parallelogram (PQ || RS and PS || QR).

Prove that the opposite angles of a parallelogram are equal.

P

R S

Q

SERIES TOPIC




10

Questions Thinking More

2. ∆KLM is an iscosceles triangle with KL KM= .

a

b

c

KN has been drawn to bisect ML. Show that ∆KMN / ∆KLN.

Show that +MNK = +LNK = 90c .

Prove that +M = +L.

3. In ∆DEF, +F = +E, if the line GD bisects +FDE. Prove ∆DEF is isosceles.

K

NLM

EG

D

F

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Similarity and Congruence Thinking MoreQuestions

4. Prove the following about the Rhombus STUV below:

a

b

c

d

∆VOS / ∆TOU

∆SOT / ∆UOV

Show that the diagonals bisect each other.

Show that the diagonals bisect at 90c .

S

T

U

O

V

SERIES TOPIC




10

Answers

Basics: Knowing More:

Using Our Knowledge:

Knowing More:

2.

1.

1.

1.

2.

3.

cmPQ 12= cmQR 18=

cmRS 20=

a

cmQR 5=

cmPT 2= cmST 4=b

P 85c+ =

S 95c+ = Q 110c+ =

D 100c+ = P 110c+ =

C 50c+ =

cmLM 10= cmAB 232=

cmKN 10=

c

d

M 45c+ = A 40c+ =

/ symbol means is congruent to. It is used to show two triangles are exactly the same in every way (corresponding sides equal and corresponding angles equal).

SSS means Side, Side, Side. It is one of the four tests that can be used to prove two triangles are congruent. If the corresponding sides of two triangles are equal, then the triangles are congruent.

SAS means Side, Angle, Side. It is one of the four tests that can be used to prove two triangles are congruent. If two sides and the included angle of two triangles are equal, then the triangles are congruent.

SAA means Side, Angle, Angle. It is one of the four tests that can be used to prove two triangles are congruent. If a corresponding side and two corresponding angles of two triangles are equal, then the triangles are congruent.

RHS means Right Angle, Hypotenuse, Side. It is one of the four tests that can be used to prove two triangles are congruent. If two right angle triangles have equal hypotenuse and an equal corresponding side, then the triangles are congruent.

||| symbol means is similar to. It is used to show two triangles have the same shape (corresponding angles are equal and corresponding sides are in proportion).

a

a

b

c

d

e

f

b

Scale factor from ∆RST to ∆UVW is 51

Scale factor from ∆UVW to ∆RST is 5

Scale factor from ∆ACD to ∆ABE is 32

Scale factor from ∆ABE to ∆ACD is 121

b cmKL 6=

c cmML 12=

b

c

cmGI 15=

cmIJ 20=

d Scale factor from ∆GFH to ∆GIJ is 221

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Similarity and Congruence Notes

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10

Notes

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Similarity and Congruence Notes

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Similarity and Congruence - intranet.cesc.vic.edu.au · Similarity and Congruence Curriculum Ready Similarity and Congruence ACMMG: 201, 220, 221, 243, 244