1B_Ch11(1). 11.1The Meaning of Congruence A Congruent Figures B Transformation and Congruence C Congruent Triangles Index 1B_Ch11(2)

1B_Ch11(1)

11.1 The Meaning of Congruence

A Congruent Figures

B Transformation and Congruence

C Congruent Triangles

Index

1B_Ch11(2)

11.2 Conditions for Triangles to be Congruent

Index

1B_Ch11(3)

Three Sides EqualA

Two Sides and Their Included Angle Equal

B

Two Angles and One Side EqualC

Two Right-angled Triangles with Equal Hypotenuses and Another Pair of Equal Sides

D

11.3 The Meaning of Similarity

A Similar Figures

B Similar Triangles

Index

1B_Ch11(4)

A Three Angles Equal

B Three Sides Proportional

Index

11.4 Conditions for Triangles to be Similar

1B_Ch11(5)

C Two Sides Proportional and their Included Angle Equal

Congruent Figures

1. Two figures having the same shape and the same size

are called congruent figures.

E.g. The figures X and Y as shown are congruent.

Index

11.1 The Meaning of Congruence 1B_Ch11(6)

Example

Index 11.1

A)

2. If two figures are congruent, then they will fit exactly

on each other.

X Y

The figure on the right shows a symmetric

figure with l being the axis of symmetry.

Find out if there are any congruent figures.

Index


Therefore, there are two congruent figures.

The line l divides the figure into 2 congruent figures,

i.e. and are congruent figures.

Find out by inspection the congruent figures among the following.

Key Concept 11.1.1

Index


A B C D

E F G H

B, D ; C, F

Transformation and Congruence

‧ When a figure is translated, rotated or reflected, the

image produced is congruent to the original figure.

When a figure is enlarged or reduced, the image

produced will NOT be congruent to the original one.

Index


Example

Index 11.1

B)

(a)(i) ____________

(ii) ____________

In each of the following pairs of figures, the red one is obtained by

transforming the blue one about the fixed point x. Determine

Index


(i) which type of transformation (translation, rotation, reflection,

enlargement, reduction) it is,

(ii) whether the two figures are congruent or not.

Reflection

Yes

(b)

(i) ____________

(ii) ____________

(c)

(i) ____________

(ii) ____________

Index


Translation

Yes

Enlargement

No

Back to Question

Key Concept 11.1.2 Index


Rotation

Yes

Reduction

No

Back to Question

(d)

(i) ____________

(ii) ____________

(e)

(i) ____________

(ii) ____________

Congruent Triangles

‧ When two triangles are congruent, all their corresponding

sides and corresponding angles are equal.

Index


Example

Index 11.1

C)

E.g. In the figure, if △ABC △XYZ, C

A B

Z

X Y

then

∠A = ∠X,

∠B = ∠Y,

∠C = ∠Z,

AB = XY,

BC = YZ,

CA = ZX.

and

Name a pair of congruent triangles in the figure.

Index


From the figure, we see that △ABC △RQP.

Given that △ABC △XYZ in the figure, find the unknowns

p, q and r.

Index


For two congruent triangles, their corresponding sides and angles are equal.

∴

∴ p = 6 cm , q = 5 cm , r = 50°

Write down the congruent triangles in each of the following.

Index


(a)

B

A

C

Y

X

Z

(a) △ABC △XYZ

(b) P

QR

S

T

U

(b) △PQR △STU

Find the unknowns (denoted by small letters) in each of the

following.

Index


(a) x = 14 ,

(a) △ABC △XYZ

1314

15A

B

Cz

x

X

YZ

z = 13

(b) △MNP △IJK

M

35°98°

47°

P

N

I

j

i

K

J

(b) j = 35° , i = 47°

Key Concept 11.1.3

Three Sides Equal

Index

11.2 Conditions for Triangles to be Congruent 1B_Ch11(18)

Example

Index 11.2

A)

‧ If AB = XY, BC = YZ and CA = ZX,

then △ABC △XYZ.

【 Reference: SSS】

A

B C

X

Y Z

Determine which pair(s) of triangles in the following are congruent.

Index

1B_Ch11(19)11.2 Conditions for Triangles to be Congruent

(I) (II) (III) (IV)

In the figure, because of SSS,

(I) and (IV) are a pair of congruent triangles;

(II) and (III) are another pair of congruent triangles.

Each of the following pairs of triangles are congruent. Which

of them are congruent because of SSS?

Index

1B_Ch11(20)

Key Concept 11.2.1


18

10

18

10A B

3

7

537

5

B

Two Sides and Their Included Angle Equal

Index


Example

Index 11.2

B)

‧ If AB = XY, ∠B = ∠Y and BC = YZ,

then △ABC △XYZ.

【 Reference: SAS】

A

B C

X

Y Z


Index


In the figure, because of SAS,

(I) and (III) are a pair of congruent triangles;

(II) and (IV) are another pair of congruent triangles.

(I) (II) (III) (IV)

In each of the following figures, equal sides and equal

angles are indicated with the same markings. Write down a

pair of congruent triangles, and give reasons.

Index


(a) (b)

(a) △ABC △CDA (SSS)

(b) △ACB △ECD (SAS)

Fulfill Exercise Objective

Identify congruent triangles from given diagram and give reasons.

Key Concept 11.2.2

Two Angles and One Side Equal

Index


Example

C)

1. If ∠A = ∠X, AB = XY and ∠B = ∠Y,

then △ABC △XYZ.

【 Reference: ASA 】

C

A B

Z

X Y

Two Angles and One Side Equal

Index


Example

Index 11.2

C)

2. If ∠A = ∠X, ∠B = ∠Y and BC = YZ,

then △ABC △XYZ.

【 Reference: AAS 】

C

A B

Z

X Y


Index


In the figure, because of ASA,

(I) and (IV) are a pair of congruent triangles;

(II) and (III) are another pair of congruent triangles.

(I) (II) (III) (IV)

In the figure, equal angles are indicated

with the same markings. Write down a

pair of congruent triangles, and give

reasons.

Index


△ABD △ACD (ASA)

Key Concept 11.2.3


Identify congruent triangles from given diagram and given reasons.


Index


In the figure, because of AAS,

(I) and (II) are a pair of congruent triangles;

(III) and (IV) are another pair of congruent

triangles.

(I) (II) (III) (IV)

In the figure, equal angles are indicated with the same

markings. Write down a pair of congruent triangles, and give

reasons.

Index

1B_Ch11(29)

Key Concept 11.2.4


A

B C

D

△ABD △CBD (AAS)

Two Right-angled Triangles with Equal Hypotenuses

and Another Pair of Equal Sides

Index


Example

Index 11.2

D)

‧ If ∠C = ∠Z = 90°, AB = XY and BC = YZ,

then △ABC △XYZ.

【 Reference: RHS】

A

B C

X

Y Z

Determine which of the following pair(s) of triangles are congruent.

Index


In the figure, because of RHS,

(I) and (III) are a pair of congruent triangles;

(II) and (IV) are another pair of congruent triangles.

(I) (II) (III) (IV)

In the figure, ∠DAB and ∠BCD are

both right angles and AD = BC.

Judge whether △ABD and △CDB

are congruent, and give reasons.

Index


Yes, △ABD △CDB (RHS)

Key Concept 11.2.5


Determine whether two given triangles are

congruent.

Similar Figures

Index

11.3 The Meaning of Similarity 1B_Ch11(33)

Example

Index 11.3

A)

1. Two figures having the same

shape are called similar figures.

The figures A and B as shown is

an example of similar figures.

2. Two congruent figures must be also similar figures.

3. When a figure is enlarged or reduced, the new figure is

similar to the original one.

Find out all the figures similar to figure A by inspection.

Index

1B_Ch11(34)

Key Concept 11.3.1

D, E


A B C D E

Similar Triangles

Index

1B_Ch11(35)

Example

Index 11.3

B)

1. If two triangles are similar, then

i. their corresponding angles are equal;

ii. their corresponding sides are

proportional.


2. In the figure, if △ABC ~ △XYZ, t

hen ∠A = ∠X, ∠B = ∠Y,

∠C = ∠Z and .ZXCA

YZBC

XYAB

A

B

C

X

Y

Z

In the figure, given that △ABC ~ △PQR,

find the unknowns x, y and z.

Index

1B_Ch11(36)

y = 98°


x = 30° , ∴

BCQR

=CARP

∴5z

=35.4

z = 535.4

= 7.5

In the figure, △ABC ~ △RPQ. Find the values of the

unknowns.

Index

1B_Ch11(37)11.3 The Meaning of Similarity

Since △ABC ~ △RPQ,

∠B = ∠P

∴ x = 90°

Index

1B_Ch11(38)11.3 The Meaning of Similarity

Also,RPAB

=PQBC

=y

39

4852

524839

= y

∴ y = 36

Also,RQAC

=PQBC

=60z

4852

486052

z =

∴ z = 65


Find the unknowns in similar triangles. Key Concept 11.3.2

Back to Question

Three Angles Equal

Index

11.4 Conditions for Triangles to be Similar 1B_Ch11(39)

Example

Index 11.4

A)

‧ If two triangles have three pairs of

equal corresponding angles, then

they must be similar.

【 Reference: AAA】

Show that △ABC and △PQR in the

figure are similar.

Index

1B_Ch11(40)11.4 Conditions for Triangles to be Similar

In △ABC and △PQR as shown,

∠B = ∠Q, ∠C = ∠R,

∠A = 180° – 35° – 75° = 70°

∠P = 180° – 35° – 75° = 70°

∴ ∠A = ∠P

∴ △ABC ~ △PQR (AAA)

∠sum of

Are the two triangles in the figure similar? Give reasons.

Index

1B_Ch11(41)

Key Concept 11.4.1


【 In △ABC, ∠B = 180° – 65° – 45° = 70°

In △PQR, ∠R = 180° – 65° – 70° = 45°】

Yes, △ABC ~ △PQR (AAA).


Identify similar triangles from given diagram and give reasons.

Three Sides Proportional

Index


Example

Index 11.4

B)

‧ If the three pairs of sides of two triangles are

proportional, then the two triangles must be similar.

【 Reference: 3 sides proportional】

a

b

c d

ef f

ceb

da

Show that △PQR and △LMN in

the figure are similar.

Index


In △PQR and △LMN as shown,

NLRP

164

41

LMPQ

82

41 ,

MNQR

123

41 ,

∴NLRP

MNQR

LMPQ

∴△PQR ~ △LMN (3 sides proportional)


Index

1B_Ch11(44)

Key Concept 11.4.2


Yes, △ABC ~ △XZY (3 sides proportional).

【 25.4

9 ZYBC

224

XZAB

, 26

12 XYAC

, 】 Fulfill Exercise Objective

Determine whether two given triangles are similar.

Two Sides Proportional and their Included Angle Equal

Index


Example

Index 11.4

C)

‧ If two pairs of sides of two triangles are proportional

and their included angles are equal, then the two

triangles are similar.

【 Reference: ratio of 2 sides, inc.∠】

x yp

qr

s

yxsq

rp ,

Show that △ABC and △FED in

the figure are similar.

Index


In △ABC and △FED as shown,

∠B = ∠E

FEAB

24 2 ,

EDBC

5.49 2

∴

EDBC

FEAB

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


Index

1B_Ch11(47)

Key Concept 11.4.3


【 ∠ ZYX = 180° – 78° – 40° = 62°, ∠ZYX = ∠CBA = 62°,

236

YZBC

, 224

XYAB

】

Yes, △ABC ~ △XYZ (ratio of 2 sides, inc. ). ∠

Documents

1B_Ch11(1). 11.1The Meaning of Congruence A Congruent Figures B Transformation and Congruence C Congruent Triangles Index 1B_Ch11(2)