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143 ISBN 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 9780170351058 144

8–01 Triangles

Classifying triangles by sides

Equilateral triangle All sides equal

60°

60° 60°

Isosceles triangle Two sides equal

Scalene triangle All sides different

×

Classifying triangles by angles

Acute-angled triangle All angles acute

Obtuse-angled triangle One angle obtuse

Right-angled triangle One 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 angle x°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 figures 145

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

Developmental Mathematics Book 4 ISBN 9780170351058 146

8–02 Quadrilaterals

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

ISBN 9780170351058 Chapter 8 Congruent and similar figures 147

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 quadrilateral d 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°

Developmental Mathematics Book 4 ISBN 9780170351058 148

Transformations8–03

WORDBANK transformation 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 3 a 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

ISBN 9780170351058 Chapter 8 Congruent and similar figures 149

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

150 Developmental Mathematics Book 4 ISBN 9780170351058

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

O 270°

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

8–04 Congruent figures

151 ISBN 9780170351058 Chapter 8 Congruent and similar figures

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

152 Developmental Mathematics Book 4 ISBN 9780170351058

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

P Q

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

B C

P Q

R

L

K

J N

M

P Q

T R

S

Pe te

r va

n Ev

er t /

A la

m y

8–05 Tests for congruent triangles

153 ISBN 9780170351058 Chapter 8 Con