Download pdf - CONGRUENT AND SIMILAR FIGURES 8web2.hunterspt-h.schools.nsw.edu.au/studentshared... · 8–05 Tests for congruent triangles 153 ISBN 9780170351058 Chapter 8 Congruent and similar

8

143ISBN 9780170351058 Chapter 8 Congruent and similar figures

IN THIS CHAPTER YOU WILL:

name and classify triangles and their properties

name and classify quadrilaterals and their side, angle and diagonal properties

solve geometry problems involving the properties and angle sums of triangles and quadrilaterals

perform transformations: translate, reflect and rotate shapes

identify congruent figures and their properties

use the ‘≡’ symbol

identify the four tests for congruent triangles: SSS, SAS, AAS, RHS

identify similar figures and their properties

use scale factors to enlarge or reduce a shape

find the length of unknown sides in similar figures

WHAT’S IN CHAPTER 8?8–01 Triangles

8–02 Quadrilaterals

8–03 Transformations

8–04 Congruent figures

8–05 Tests for congruent triangles

8–06 Similar figures

8–07 Enlargements and reductions

8–08 Finding sides in similar figures

CONGRUENT AND SIMILAR FIGURES

Shutterstock.com/Ratikova

Developmental Mathematics Book 4 ISBN 9780170351058144

8–01 Triangles

Classifying triangles by sides

Equilateral triangleAll sides equal

60°

60° 60°

Isosceles triangleTwo sides equal

Scalene triangleAll sides different

×

Classifying triangles by angles

Acute-angled triangleAll angles acute

Obtuse-angled triangleOne angle obtuse

Right-angled triangleOne right angle

The angle sum of any triangle is 180°. The exterior angle of a triangle is the sum of the two interior opposite angles.

b°

a° c°

b°

exterior anglex°a°

a + b + c = 180 x = a + b

EXAMPLE 1

Find the value of each pronumeral, giving reasons. a

41°

x°68°

b

n°

44° c

52°

46°

m°

SOLUTION

a 68 + 41 + x = 180 (angle sum of a triangle) x = 180 – 68 – 41 x = 71

b 44 + 44 + n = 180 (angle sum of an isosceles triangle) n = 180 – 44 – 44 n = 92

c m = 52 + 46 (exterior angle of a triangle) m = 98

exercise 8–01

ISBN 9780170351058 Chapter 8 Congruent and similar figures145

1 What type of triangle has three equal sides? Select the correct answer A, B, C or D.

A equilateral B right-angled C scalene D isosceles

2 In an isosceles triangle, where are the equal angles? Select A, B, C or D.

A All three angles are equal B At the bottom of the triangle

C At the top of the triangle D Opposite the equal sides

3 Classify each triangle according to its sides.a b c

4 Classify each triangle in Question 3 according to its angles.

5 Find the value of each pronumeral.

a

16°

x° y°

b m°

o°

n° c

w°

v°

6 For each triangle, name:

i the exterior angle ii the two interior opposite angles

a f°

d° e° c°

b s°

p°

r°t°

7 Find the value of each pronumeral, giving reasons. a

x°

55° 46°

b 63°

n°52°

c 48°

34°

n°

d

78°

a°

e x°

34°

f n°

42°

8 Sketch each triangle described.

a Isosceles and right-angled b Scalene and obtuse-angled



Convex quadrilateral

All vertices point outwards.All diagonals lie inside the shape.

Non-convex quadrilateral

One vertex points inwards. One diagonal lies outside the shape.

Parallelogram

Trapezium Rectangle Rhombus

Square Kite

EXAMPLE 2

Find the value of each pronumeral, giving reasons.

a 64°

x°

b 114°

v°

c n°

121°

34°SOLUTION

a x + 64 = 180 (co-interior angles on parallel lines) x = 180 – 64 x = 116

b v = 114 (opposite angles of a parallelogram)

c • = 121 (opposite angles of a kite) 121 + 121 + 34 + n = 360 (angle sum of a quadrilateral) n = 360 – 121 – 121 – 34 n = 84

The angle sum of any quadrilateral is 360°.a + b + c + d = 360

a°

b° c°

d°

exercise 8–02


1 Which quadrilateral has equal diagonals? Select the correct answer A, B, C or D.

A rhombus B rectangle C parallelogram D trapezium

2 What is the size of all angles in a square? Select A, B, C or D.

A 180° B 90° C 360° D 60°

3 Draw a sketch of each quadrilateral.a trapezium b kite c non-convex quadrilaterald square e convex quadrilateral f parallelogram

4 Name each quadrilateral.a b c

5 Find the value of each pronumeral, giving reasons.

a 127°

122°

x°

b

p°

c

93°

48°

n°

d 62°

x°

e 124°

v°

f n°

118°

32°

g

n°

h

78° n°

i 68° x°


Transformations8–03

WORDBANKtransformation The process of moving or changing a shape, by translation, reflection or rotation.

translation The process of ‘sliding’ a shape, moving it up, down, left or right.

reflection The process of ‘flipping’ a shape, making it back-to-front like in a mirror.

rotation The process of ‘spinning’ a shape, around a point, tilting it sideways or upside-down

EXAMPLE 3a This L-shape has been translated 3 units to the right.

b This L-shape has been reflected across the line AB.

A

B

c This L-shape has been rotated 90° clockwise about point O.

O

exercise 8–03


1 What type of transformation occurs if a shape is turned anticlockwise? Select the correct answer A, B, C or D. A translation B reflection C transformation D rotation

2 What type of transformation occurs if a shape is flipped over a line? Select A, B, C or D.A translation B reflection C transformation D rotation

3 Copy this rhombus on grid paper and perform each translation to it.a 2 units right b 2 units up c 1 unit left d 1 unit down

4 Copy each shape on grid paper in the line drawn.

exercise 8–03

150Developmental Mathematics Book 4 ISBN 9780170351058

5 Copy this diagram onto grid paper and perform each rotation.a 90° clockwise about O b 180° about Pc 90° anticlockwise about P d 270° clockwise about O

OP

6 Copy each diagram onto grid paper and perform each rotation about point O. a

O270°

b

O

90°

7 Copy each shape onto square dot paper and translate it in the direction and distance indicated by the arrow.a A

B O

b



These two triangles are congruent.A

B C

Y Z

X

∠A matches ∠BX, ∠B matches ∠Z , ∠C matches ∠Y .We can write ‘ ≡ABC XZY ’ in matching order of the vertices, where ‘≡’ stands for ‘is congruent to’.Congruent figures can be formed by translation, reflection, rotation or a combination of them.

EXAMPLE 4

These two triangles are congruent.A

B C

X

Y Z

a Which transformation turns the first triangle into the second triangle?

b List the three pairs of matching angles.

c Copy and complete this congruence statement: ≡ABC ___

SOLUTION

a Reflection

b ∠A = ∠X, ∠B = ∠Z, ∠C = ∠Y

c ≡ABC XZY matching order of vertices

Congruent figures are shapes that are identical in every way, with the same shape and the same size. Matching sides are equal Matching angles are equal

exercise 8–04


1 If these two quadrilaterals are congruent, which angle in PQRS matches ∠A? Select the correct answer A, B, C or D.

A B

C

D

PQ

S R

A P B Q C R D S

2 For Question 1, which is the correct order of vertices to complete the congruence statement ‘ABCD ≡ _____’? Select A, B, C or D.

A PQRS B QSRP C RSPQ D SRQP

3 Find three pairs of congruent figures and state which transformation can be used between them.A B C D

E F G H

4 If ≡ABC RPQ, list all pairs of matching sides and matching angles.

5 These two pentagons are congruent.

a List all pairs of matching angles.

b Copy and complete: JKLMN ≡ _________.

6 The Dutch artist Maurits Cornelius Escher used congruent shapes to draw amazing works of art. Describe the congruent shapes in this Escher drawing. Which transformation has he used?

A

BC

PQ

R

L

K

J N

M

P Q

T R

S

Pete

r va

n Ev

ert /

Ala

my



There are 4 tests to prove that two triangles are congruent.1. SSS Three sides of one triangle are equal to three sides of the other triangle.

2. SAS Two sides and the included angle of one triangle equal two sides and the included angle of the other

triangle.

3. AAS Two angles and one side of one triangle equal two angles and the matching side of the other triangle.

*

*

4. RHS A right angle, hypotenuse and a side of one triangle equal to the right angle, hypotenuse and the matching side of the other triangle.

EXAMPLE 5

Which two triangles are congruent? Which congruence test proves this?

A

B C

P Q

R

Y

X

Z

SOLUTION

≡ABC RQP Test used was AAS.

XYZ is not congruent to these triangles because YZ is not the matching side.

The included angle means the angle in between the two sides.

exercise 8–05


1 Which test proves that these triangles are congruent? Select the correct answer A, B, C or D.X

Y Z

P

RQ

A SAS B RHS C SSS D AAS

2 For Question 1, XYZ ≡ _____? Select A, B, C or D.

A PQR B RQP C RPQ D PRQ

3 Find three pairs of congruent triangles and state which test you used.

A 5

13 B

3 3

4

C

15

D 4

3 3

E

15

F 513

4 Draw two congruent triangles that follow:a the SAS test b the RHS test c the AAS test

5 Find three pairs of congruent triangles. Remember to write the vertices in matching order and state which test you used.

A

B C

106

P

R6

Q9

9

W

X Y

8

3

9

6E

D

F

9610

I

HG

M

O

83

N



These two quadrilaterals are similar.

GF

ED

M

H

K

L

∠G = ∠M, ∠F = ∠L, ∠E = ∠K, ∠D = ∠H.Each side of MLKH is exactly double the length of the matching side in GFED.We can write ‘GFED ||| MLKH’ in matching order of the vertices, where ‘|||’ stands for ‘is similar to’.

EXAMPLE 6

Find pairs of similar figures.

A

B

C

D

E

SOLUTION

A and C, B and E. They are the same shape but different sizes.

Similar figures have the same shape but not necessarily the same size. One is an enlargement (or reduction) of the other.

Matching angles are equal Matching sides have lengths that are in the same ratio

Shut

ters

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.com

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exercise 8–06


1 Which phrase is correct about similar figures? Select the correct answer A, B, C or D.

A same shape, same size B same shape, different size

C different shape, same size D different shape, different size

2 Which pair of shapes are always similar? Select A, B, C or D.

A any two triangles B any two rectangles

C any two parallelograms D any two squares

3 Which set of shapes include similar figures?A B

C

D E BUSSTOP

4 Find 7 pairs of similar figures.

A BC

D

J

E FG

L

H

M

Q

N

O P R

I

K

exercise 8–06


5 Name two pairs of similar triangles. Remember to write the vertices in matching order.A

B

C

70°

20°

D

E

F

70°

70°

40°

X Y

Z

70°

20°

P

Q R

40°

70° 70°

6 Two pairs of matching angles are equal in these two triangles.

75°35°

A

B C 35° 75°

P

Q R

a Find the size of ∠A and ∠P.

b Is angle A the same size as angle P?

c Is ABC ||| PQR?

7

A

B C30°

D

E30° F

a What are the sizes of angles A and D?

b Are the two triangles similar?

c Name pairs of matching sides.

d Complete: DEF ||| ________.

Enlargements and reductions8–07


EXAMPLE 7

Find the scale factor for each enlargement or reduction.

a

2 3

original image10

64

5

b

original image

3 2

SOLUTION

a Scale factor = 42

(or 63

105

or ) image lengthoriginal length

= 2

The image is double the size of the original: an enlargement.

b Scale factor = 23 image length

original length

The image is 23

the size of the original: a reduction.

WORDBANKenlargement A shape or diagram that has been made larger in size.

reduction A shape or diagram that has been made smaller in size.

image A transformed shape or diagram after it has been enlarged or reduced.

scale factor The amount the original figure is multiplied by to create an enlargement or reduction.

Scale factor = image length

original length If the scale factor is greater than 1, then the image is an enlargement. If the scale factor is between 0 and 1, then the image is a reduction.

exercise 8–07


1 If the original figure had a height of 4 cm and the image is 3 cm high, what is the scale factor? Select the correct answer A, B, C or D.

A 43

B 12 C 34

D 1.3

2 In an enlargement with scale factor 1.5, the original figure is 12 cm long. What is the length of the image? Select A, B, C or D.

A 8 cm B 16 cm C 24 cm D 18 cm

3 Find the scale factor for each enlargement or reduction.

a

2 4

10

84

5

b

2 3

c

3

5

d

4 2

e

2.5

7.5

f 9

3

exercise 8–07


g

2

3

15

10

h 1.2

0.9

4 Enlarge each figure using a scale factor of:i 2 ii 1.5a b c

5 Reduce each figure using a scale factor of:

i 12

ii 0.8

a b c

6 a Enlarge ABC using a scale factor of 2.5.

3 cm

4 cm

A

B C

b Reduce ABC using a scale factor of 23

.



EXAMPLE 8

Find the value of each pronumeral in this pair of similar triangles.

3 cm 4 cm

y cm

4.5 cm w cm

7.5 cm

SOLUTION

First calculate the scale factor by comparing matching sides 3 cm and 4.5 cm.

Scale factor = 4 53.

= 1.5

\ w = 4 × 1.5 4 and w are matching sides. = 6

\ y = 7.5 ÷ 1.5 y and 7.5 are matching sides. = 5

From the diagram, a length of 6 cm looks reasonable for w.

From the diagram, a length of 5 cm looks reasonable for y.

Get

ty im

ages

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Sund

exercise 8–08


1 Which phrase is true about matching sides in similar figures? Select the correct answer A, B, C or D.

A equal B twice the original

C in the same ratio D random

2 Which phrase is true about matching angles in similar figures? Select A, B, C or D.

A equal B twice the original

C in the same ratio D random

3 a Copy and complete this table of measurements for the diagram.

Length Height Door length Door height

Small house 4 units

Big house

Scale factor

hei

ght

hei

ght

b Are the houses similar figures?

4 Find the value of the pronumeral in each pair of similar figures.

a

2 cm

3 cm

6 cm

x cm

b 4 mm

6 mm

w mm

9 mm

c

y m

2.5 m

7.5 m

9 m d

2 m

1.5 m

3.75 m

x m

exercise 8–08


5 In this diagram, ∠A = ∠D and ABC ||| DEC. Find the length of AC.

1.5 cm2.5 cm

3 cm C E

D

B

A

A = D∠∠

6 a Copy and complete: ABC ||| ______.

A

P

B C

4

3

5

8

6Q R

x

b Show that ABC is a right-angled triangle.

c Find the value of x using:

i similar figures ii Pythagoras’ theorem

Shut

ters

tock

.com

/DT1

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LANGUAGE ACTIVITY

164

WORD SCRAMBLEUnscramble the letters below to form words used in this chapter.

LANGUAGE ACTIVITY

Part A General topicsCalculators are not allowed.

1 Evaluate 23

15

− .

2 Factorise 6xy2 + 2y.

3 Find the mean of 1, 2, 8, 3, 6.

4 Ruchi pays a grocery bill of $83.45 with a $100 note. Calculate the change.

5 Use three letters to name the marked angle.

Q

PR

6 What type of angle is marked in Question 5?

7 Write Pythagoras’ theorem for this triangle.

5

12

x

8 Find the value of x in Question 7.

9 Simplify 123

2

2

a bb

10 Write a simplified algebraic expression for the perimeter of this rectangle.

5r

3t

PRACTICE TEST 8

1 IMISARL

2 GLANTIRE

3 TRONCUGEN

4 GERIFU

5 SLANTRITOAN

6 TREDICOUN

7 TALAQUILRIDAR

8 GERMLENAENT

9 FLEIONTECR

10 MATAFORSNORNIT

11 SETT

12 TORNAITO

13 AMIGE

14 GLANE

15 TIMGCAHN



PRACTICE TEST 8

Part B Congruent and similar figuresCalculators are allowed.

8–01 Triangles

11 What type of triangle is this? Select the correct answer A, B, C or D.

A acute-angled and isosceles B acute-angled and scaleneC right-angled and isosceles D right-angled and scalene

12 Find the value of x. Select A, B, C or D.

36° x°

A 72° B 36° C 108° D 180°


13 Name each quadrilateral. a b

14 Find the value of each pronumeral, giving reasons. a

85°

107°

124°

x°

b m°

82°


PRACTICE TEST 8

8–03 Transformations

15 Name the type of transformation that has been performed on quadrilateral ABCD to create the congruent figure on the right.

B

CD

A

R

S

U

T


16 Copy and complete for the quadrilaterals in Question 15: ABCD ≡ ______


17 Which test proves that these triangles are congruent?

20

16

20

16


18 Which two figures are similar? A B C D

8–07 Enlargements and reductions

19 Find the scale factor for each pair of similar figures.a

59

18

15

6

3

b

3 4.5


20 Find the value of x in these similar figures.

4

2.5

10

x