properties of congruent triangles

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Properties of Congruent Triangles

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Properties of Congruent Triangles. Congruence. Are the following pairs of figures the same?. They are the same!. Figures having the same shape and size are called congruent figures. Congruent Triangles and their Properties. - PowerPoint PPT Presentation

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Page 1: Properties of Congruent Triangles

Properties of Congruent Triangles

Page 2: Properties of Congruent Triangles

Figures having the same shape and size are called congruent figures.

Are the following pairs of figures the same?

Congruence

They are the same!

Page 3: Properties of Congruent Triangles

If two triangles have the same shape and size, they are called congruent triangles.

Congruent Triangles and their Properties

A X

B YC Z

For the congruent triangles △ABC and △XYZ above,

A = X, B = Y, C = Z

AB = XY, BC = YZ, CA = ZX

Page 4: Properties of Congruent Triangles

A and X, B and Y, C and Z are called

AB and XY, BC and YZ, CA and ZX are called

A and X, B and Y, C and Z are called

A X

B YC Z

corresponding vertices.

correspondingsides.

correspondingangles.

Page 5: Properties of Congruent Triangles

is congruent to △XYZ△ABCThe corresponding vertices of congruent triangles should be written in the same order.

In the above example,we can also write △BAC △YXZ, but NOT △CBA △XYZ.

A X

B YC Z

The properties of congruent triangles are as follows:

AB = XY, BC = YZ,

A =X, B = Y, C = Z CA = ZX(ii) All their corresponding sides are equal.

(i) All their corresponding angles are equal,

Page 6: Properties of Congruent Triangles

If △ABC △XYZ …

XY

Z

4 cm

A

B

C

4.5 cm

40°

According to the properties of congruent triangles,

AB = 4.5 cm AC = 4 cm

B = 40°

△ △ A B C X Y Z

XY XZ

Y

= =

=

Page 7: Properties of Congruent Triangles

In the figure, △ABC △ PQR. Find the unknowns.

Follow-up question 1

A

B

C130° 30°

x cm

4 cm

7 cm

P

Q

Ry9 cm

z cm

According to the properties of congruent triangles,AC PR

CBy 180 30130180

20

4zBCQR

9x

AP

Page 8: Properties of Congruent Triangles

Example 1

In the figure, AB = 5 cm, AC = 4 cm and BC = 7 cm. If △ABC △DFE, find DE, EF and DF.

Solution

According to the properties of congruent triangles,

cm 5

cm 7

cm 4

ABDF

BCEF

ACDE

Page 9: Properties of Congruent Triangles

Example 2

In the figure, AB = 7 cm, ∠A = 50° and ∠B = 30°. If △ABC △PRQ, find PR, ∠P and ∠R.

Solution

According to the properties of congruent triangles,

30

50

cm 7

BR

AP

ABPR

Page 10: Properties of Congruent Triangles

Conditions for Congruent Triangles

Page 11: Properties of Congruent Triangles

Yes, because AB = XY, BC = YZ, CA = ZX,

∠A =∠X, ∠B =∠Y, ∠C =∠Z.

But, can we say that two triangles are congruent when only some of the properties of congruence are satisfied?

Are these two triangles congruent?

A

CB

X

Y Z

Yes… let’s see the following 5 conditions for congruent triangles first.

Page 12: Properties of Congruent Triangles

Condition I SSS

In △ABC and △XYZ,

if AB = XY, BC = YZ and CA = ZX,

then △ABC △XYZ.

[Abbreviation: SSS]

C

A BZ

X Y

Page 13: Properties of Congruent Triangles

For example,

T

U

V

2 cm

4 cm

5 cm

D

E

F 2 cm

5 cm

4 cm

TU = FE, UV = ED, TV = FD

△TUV △FED (SSS)

Page 14: Properties of Congruent Triangles

In △ABC and △XYZ,

if AB = XY, AC = XZ and A = X,

then △ABC △XYZ.

[Abbreviation: SAS]

Condition II SAS

C

A BZ

XY

Note that ∠A and ∠X are the included angles of the 2 given sides.

Page 15: Properties of Congruent Triangles

For example,

UV = FE, TV = DE, V = E

△TUV △DFE (SAS)

T

U

V

120°2 cm

2.5 cm

D

E

F2.5 cm

2 cm120°

Page 16: Properties of Congruent Triangles

Follow-up question 2Determine whether each of the following pairs of triangles are congruent and give the reason.

(a)A

B

C

E

G

F

3 cm

3.2 cm

3 cm

3.2 cm

2.5 cm

2.5 cm

△ABC △EGF (SSS) ◄ AB = EG, BC = GF, AC = EF

Page 17: Properties of Congruent Triangles

(b)I

J

K

2 cm

2.5 cm50°

N

M

L

2 cm

2.5 cm

50°

△IJK △MNL (SAS) ◄ IJ = MN, ∠J = ∠N, JK = NL

Page 18: Properties of Congruent Triangles

Example 3

Are △MNP and △YZX in the figure congruent? If they are, give the reason.

Yes, △MNP △YZX. (SSS)

Solution

Page 19: Properties of Congruent Triangles

Example 4

In the figure, WX = WY and ZX = ZY. Are △WXZ and △WYZ congruent? If they are, give the reason.

Solution

Yes, △WXZ △WYZ. (SSS)

Page 20: Properties of Congruent Triangles

Example 5

Which two of the following triangles are congruent? Give the reason.

Solution

△PQR △WUV (SAS)

Page 21: Properties of Congruent Triangles

Example 6

In the figure, AB = CD = 8 cm and ∠ABD = ∠CDB = 30°. Are △ABD and △CDB congruent? If they are, give the reason.

Yes, △ABD △CDB. (SAS)

Solution

Page 22: Properties of Congruent Triangles

Example 7

Which two of the following triangles are congruent? Give the reason.

Solution△DEF △ZYX (ASA)

Page 23: Properties of Congruent Triangles

In △ABC and △XYZ,

if A = X, B = Y and AB = XY,

then △ABC △XYZ.

[Abbreviation: ASA]

Condition III ASA

C

A B

Z

X Y

Note that AB and XY are the included sides of the 2 given angles.

Page 24: Properties of Congruent Triangles

U

T

V

130°20°

4 cm

D

E

F

130°20°

4 cm

For example,

U = F, UV = FD, V = D

△TUV △EFD (ASA)

Page 25: Properties of Congruent Triangles

Condition IV AAS

In △ABC and △XYZ,

if A = X, B = Y and AC = XZ,

then △ABC △XYZ.

[Abbreviation: AAS]

C

A B

Z

X Y

Note that AC and XZ are the non-included sides of the 2 given angles.

Page 26: Properties of Congruent Triangles

T

U V

D

E

F

130°

20°

20°7 cm

7 cm

For example,

U = F, TV = EDV = D,

△TUV △EFD (AAS)

130°

Page 27: Properties of Congruent Triangles

Follow-up question 3In each of the following, name a pair of congruent triangles and give the reason.

(a) A

BC

E

F

G

45°

40°

40°

45°

5.25 cm

5.25 cm

△ABC △FEG (ASA) ◄ ∠B = ∠E, BC = EG, ∠C = ∠G

Page 28: Properties of Congruent Triangles

(b)

△IJK △MNL (AAS)

K

IL

J

M

N

100°

20°

20°100°

12 cm 12 cm

B

A

C100°

12 cm

20°

◄ ∠J = ∠N, ∠K = ∠L, IK = ML

Page 29: Properties of Congruent Triangles

Example 8

In the figure, ∠BAD = ∠CAD and AD⊥BC. Are △ABD and △ACD congruent? If they are, give the reason.

Solution

Yes, △ABD △ACD. (ASA)

Page 30: Properties of Congruent Triangles

Example 9

Which two of the following triangles are congruent? Give the reason.

Solution

△PQR △ZYX (AAS)

Page 31: Properties of Congruent Triangles

Example 10

In the figure, ∠ABC = ∠CDA and ∠ACB = ∠CAD. Are △ABC and △CDA congruent? If they are, give the reason.

Solution

Yes, △ABC △CDA. (AAS)

Page 32: Properties of Congruent Triangles

In △ABC and △XYZ,

if C = Z = 90°, AB = XY and

BC = YZ (or AC = XZ),

then △ABC △XYZ.

[Abbreviation: RHS]

Condition V RHS

A

B C

X

Y Z

Page 33: Properties of Congruent Triangles

For example,

2 cm 2 cm 5 cm5 cm

T

U V

D

E

F

U = F = 90°, TV = ED, TU = EF

△TUV △EFD (RHS)

Page 34: Properties of Congruent Triangles

Are there any congruent triangles? Give the reason.

Follow-up question 4

A

B C

D

6 cm

6 cm

Yes, △ABC △ADC. (RHS) ◄ ∠B = ∠D = 90°, AC = AC, BC = DC

Page 35: Properties of Congruent Triangles

C

A B

Z

X Y

1. SSS

C

A B

Z

X Y

2. SAS

C

A B

Z

X Y

3. ASA

C

A B

Z

X Y

4. AAS

A

B C

X

Y Z

5. RHS

To sum up, two triangles are said to be congruent if any ONE of the following FIVE conditions is satisfied.

Page 36: Properties of Congruent Triangles

Example 11

Are △ABC, △RPQ and △XYZ in the figure congruent? If they are, give the reasons.

Solution

△ABC △RPQ (RHS)△XYZ △RPQ (SAS)∴ △ABC, △RPQ and △XYZ are congruent.

Page 37: Properties of Congruent Triangles

Example 12

In the figure, AB⊥BC, DC⊥BC and AC = DB. Are △ABC and △DCB congruent? If they are, give the reason.

Solution

Yes, △ABC △DCB. (RHS)

Page 38: Properties of Congruent Triangles

Properties of Similar Triangles

Page 39: Properties of Congruent Triangles

Similar figures have the same shape but not necessarily

the same size.

The following pairs of figures have the same shape, they are called similar figures.

Similarity

Page 40: Properties of Congruent Triangles

Similar Triangles and their Properties

If two triangles have the same shape, they are called similar triangles.

For the similar figures △ABC and △XYZ above,

A = X, B = Y, C = Z

AB XY

= BC YZ

CA ZX

=

AX

B YC Z

Page 41: Properties of Congruent Triangles

A

X

B YC Z

A and X, B and Y, C and Z are called

AB and XY, BC and YZ, CA and ZX are called

A and X, B and Y, C and Z are called corresponding vertices.

correspondingsides.

correspondingangles.

Page 42: Properties of Congruent Triangles

AX

B YC Z

The properties of similar triangles are as follows:

A = X, B = Y, C = Z

AB XY

= BC YZ

CA ZX

=(ii) All their corresponding sides are proportional.

(i) All their corresponding angles are equal,

~is similar to △XYZ△ABCNote: The corresponding vertices of congruent triangles should be written in the same order.

Page 43: Properties of Congruent Triangles

If △ABC ~ △XYZ ...

4 cm

A

B

C

4.5 cm

40°

XY

Z

2 cm

According to the properties of similar triangles,

BY 40 AC

XZABXY

cm 4cm 2

cm 5.4XY

cm 25.2 △ ~ △ A B C X Y Z

◄ XZ and AC are corresponding sides.

Page 44: Properties of Congruent Triangles

In the figure, △ABC ~ △PQR. Find the unknowns.

Follow-up question 5

A

B

C132° 25°

10 cm

4 cm

x cm

According to the properties of similar triangles,

P

Q

Ry5 cm

z cm4 cm

PRAC

PQAB

510

4x

8x2z

AP CBy 180

25132180 23

ACPR

BCQR

105

4z

Page 45: Properties of Congruent Triangles

Example 13

In △ABC and △RQP, BC = 1 cm, PQ = 2 cm, QR = 5 cm and PR = 4 cm. If △ABC ~ △RQP, find AB and AC.

Page 46: Properties of Congruent Triangles

Solution

According to the properties of similar triangles,

cm 5.2cm 2

cm 1

cm 5

AB

AB

QP

BC

RQ

AB

cm 2cm 2

cm 1

cm 4

AC

AC

QP

BC

RP

AC

Page 47: Properties of Congruent Triangles

Example 14

In the figure, AD = 3 cm, AC = 2 cm, CE = 4 cm, ∠A = 60° and BC⊥AE. If △ABC ~ △AED, find ∠E and AB.

Page 48: Properties of Congruent Triangles

SolutionAccording to the properties of similar triangles,

ADE = ACB = 90 In △ ADE,

∵ 180EADEA

30

1809060

E

E

cm 4

cm 3

cm 2

cm )42(

AB

ABAD

AC

AE

AB

Page 49: Properties of Congruent Triangles

Conditions for Similar Triangles

Page 50: Properties of Congruent Triangles

Conditions for Similar Triangles

(i) All their corresponding angles are equal.

(ii)All their corresponding sides are proportional.

A

B CX

Y Z

We have learnt that if two triangles are similar, then

Two triangles are similar if any one of the following

three conditions is satisfied.

Page 51: Properties of Congruent Triangles

In △ABC and △XYZ,

if A = X, B = Y and C = Z,

then △ABC ~ △XYZ.

[Abbreviation: AAA]

Condition I AAA

A

B C

X

YZ

Page 52: Properties of Congruent Triangles

U = F, T = E, V = D

△TUV ~ △EFD (AAA)

For example,

T

U V

127° 25°

28°D

E

F

127°25°

28°

Page 53: Properties of Congruent Triangles

Condition II 3 sides prop.

In △ABC and △XYZ,

if

then △ABC ~ △XYZ.

[Abbreviation: 3 sides prop.]

,ZXCA

YZBC

XYAB

A

B C

X

Y Z

Page 54: Properties of Congruent Triangles

T

U

V

4 cm

2 cm

3 cm

D

E

F1.5 cm

1 cm

2 cm

For example,

△TUV ~ △DFE (3 sides prop.)

UV 2 cm FE 1 cm

= = 2, TV 3 cm DE 1.5 cm

= = 2, TU 4 cm DF 2 cm

= = 2

Page 55: Properties of Congruent Triangles

Condition III ratio of 2 sides, inc.

A

B C

In △ABC and △XYZ,

if and B = Y,

then △ABC ~ △XYZ.

[Abbreviation: ratio of 2 sides, inc. ]

YZBC

XYAB

X

Y Z

Page 56: Properties of Congruent Triangles

For example,

D

E

F

120°

2 cm

1.5 cm

V4 cm

U

T

120°3 cm

△TUV ~ △EFD (ratio of 2 sides, inc. )

UV 4 cm FD 2 cm

= = 2, UT 3 cm FE 1.5 cm

= = 2, U = F

Page 57: Properties of Congruent Triangles

Follow-up question 6Determine whether each of the following pairs of triangles are similar and give the reason.

(a)

E

G

F

2.4 cm

2.8 cm

2 cm

A

B

C

3 cm

3.5 cm

2.5 cm

△ABC ~ △EGF (3 sides prop.) ABEG

BCGF

◄ =ACEF

=

Page 58: Properties of Congruent Triangles

(b)

△ABC ~ △ZXY (AAA)

C

A B

95°

40° 45°

45°X

Y

Z40°

95°

I

J

K

2.4 cm

3 cm

50°

N

M

L

1.6 cm

2 cm

50°

△IJK ~ △MNL (ratio of 2 sides, inc. )

(c)

◄ ∠A = ∠Z, ∠B = ∠X, ∠C = ∠Y

IJMN

JKNL

◄ = , ∠J = ∠N

Page 59: Properties of Congruent Triangles

Example 15

Are △ABC and △XZY in the figure similar? If they are, give the reason.

Yes, △ABC ~ △XZY. (AAA)

Solution

Page 60: Properties of Congruent Triangles

Example 16

Are △ABC and △QRP in the figure similar? If they are,give the reason.

Solution

485082180

824850180

R

C

∴ △ ABC ~ △ QRP (AAA)

Page 61: Properties of Congruent Triangles

Example 17Which two of the following triangles are similar? Give thereason.

Solution

2

1

cm 10

cm 52

1

cm 8

cm 4

2

1

cm 6

cm 3

PQ

CARP

BC

QR

AB

∴ △ ABC ~ △ QRP (3 sides prop.)

Page 62: Properties of Congruent Triangles

Example 18In the figure, AB = 15 cm, BC = 12 cm, AC = 9 cm, BD = 20 cm and CD = 16 cm. Are △ABC and △BDC similar? If they are, give the reason.

Solution

4

3

cm 12

cm 94

3

cm 16

cm 124

3

cm 20

cm 51

CB

CADC

BCBD

AB

∴ △ ABC ~ △ BDC (3 sides prop.)

Page 63: Properties of Congruent Triangles

Example 19

Which two of the following triangles are similar? Give the reason.

Page 64: Properties of Congruent Triangles

3

2

cm 7.5

cm 5

3

2

cm 5.4

cm 3

RQ

AC

PQ

BC

∴ △ ABC ~ △ RPQ (ratio of 2 sides, inc. )

Solution

3

2

cm 7.5

cm 5

3

2

cm 5.4

cm 3

RQ

AC

PQ

BC

∴ △ ABC ~ △ RPQ (ratio of 2 sides, inc. )

Page 65: Properties of Congruent Triangles

Example 20

In the figure, PT = 2.7 cm, TR = 3.3 cm, QR = 3 cm, TS = 4.8 cm, RS = 2.4 cm and ∠PRQ = ∠TSR. Are △PQR and △TRS similar? If they are, give the reason.

4

5

cm 2.4

cm 34

5

cm 8.4

cm )3.37.2(

RS

QRTS

PR

∴ △ PQR ~ △ TRS (ratio of 2 sides, inc. )

Solution

Page 66: Properties of Congruent Triangles

Example 2 (Extra)

In the figure, △ABC △EDF and △FED △IHG. Find GH, HI and IG.

According to the properties of congruent triangles,

cm 21

cm 22

cm 20

CBFDIG

ACEFHI

BADEGH

Solution

Page 67: Properties of Congruent Triangles

Example 10 (Extra)

In the figure, AB = PQ, ∠ABC = ∠QRP and ∠ACB = ∠PQR. Are △ABC and △QRP congruent? If they are, give the reason.

Solution

Cannot be determined.Since the length of PR may not equal to AB.

Page 68: Properties of Congruent Triangles

Example 12 (Extra)In the figure, AGE, CGF and BCD are straight lines.

(a) Are △ABC and △CDE congruent? If they are, give the reason.

(b) Are △FAC and △FEC congruent? If they are, give the reason.

(a) Yes, △ ABC △ CDE. (SAS)

(b) According to the properties of congruent triangles,

AC CE

Yes, △ FAC △ FEC. (RHS)

Solution(a) Yes, △ ABC △ CDE. (SAS)

(b) According to the properties of congruent triangles,

AC CE

Yes, △ FAC △ FEC. (RHS)

(a) Yes, △ ABC △ CDE. (SAS)

(b) According to the properties of congruent triangles,

AC CE

Yes, △ FAC △ FEC. (RHS)

(a) Yes, △ ABC △ CDE. (SAS)

(b) According to the properties of congruent triangles,

AC CE

Yes, △ FAC △ FEC. (RHS)

Page 69: Properties of Congruent Triangles

Example 20 (Extra)

In the figure, KN = 6 cm, NM = 5 cm, LM = 4

33 cm, KM =

2

14 cm and ∠KML

= ∠KNM.

(a) Name a pair of similar triangles in the figure and give the reason.

(b) Hence, find the value of z.

In the figure, KN = 6 cm, NM = 5 cm, LM = 4

33 cm, KM =

2

14 cm and ∠KML

= ∠KNM.

(a) Name a pair of similar triangles in the figure and give the reason.

(b) Hence, find the value of z.

In the figure, KN = 6 cm, NM = 5 cm, LM = 4

33 cm, KM =

2

14 cm and ∠KML

= ∠KNM.

(a) Name a pair of similar triangles in the figure and give the reason.

(b) Hence, find the value of z.

In the figure, KN = 6 cm, NM = 5 cm, LM = 4

33 cm, KM =

2

14 cm and ∠KML

= ∠KNM.

(a) Name a pair of similar triangles in the figure and give the reason.

(b) Hence, find the value of z.

In the figure, KN = 6 cm, NM = 5 cm, LM = 4

33 cm, KM =

2

14 cm and ∠KML

= ∠KNM.

(a) Name a pair of similar triangles in the figure and give the reason.

(b) Hence, find the value of z.

Page 70: Properties of Congruent Triangles

Solution

(a)

4

3

cm 6

cm 2

14

4

3

cm 5

cm 4

33

NK

MK

MN

LM

∴ △ KLM ~ △ KMN (ratio of 2 sides, inc. )

(a)

4

3

cm 6

cm 2

14

4

3

cm 5

cm 4

33

NK

MK

MN

LM

∴ △ KLM ~ △ KMN (ratio of 2 sides, inc. )

Page 71: Properties of Congruent Triangles

(b) According to the properties of similar triangles,

8

33

cm 6

cm 2

14

cm 2

14

cm

z

z

NK

MK

KM

KL

(b) According to the properties of similar triangles,

8

33

cm 6

cm 2

14

cm 2

14

cm

z

z

NK

MK

KM

KL

(b) According to the properties of similar triangles,

8

33

cm 6

cm 2

14

cm 2

14

cm

z

z

NK

MK

KM

KL

(b) According to the properties of similar triangles,

8

33

cm 6

cm 2

14

cm 2

14

cm

z

z

NK

MK

KM

KL