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Plane Geometry

Mathletics Instant

Workbooks

Student Book - Series L-2

x2 + y2 = r2

yr

x

Plane geometryStudent Book - Series L 2

Contents

Topics

Practice Tests

Topic 1 - Notation, symbols, terms and definitions __ /__ /__

Topic 1 - Topic test A __ /__ /__

Topic 2 - Angles __ /__ /__

Topic 2 - Topic test B __ /__ /__

Topic 3 - Corresponding, alternate and co-interior angles __ /__ /__Topic 4 - Angle sum of a triangle and the exterior angle of a triangle __ /__ /__Topic 5 - Angle sum of a quadrilateral __ /__ /__Topic 6 - Angle sum of a polygon __ /__ /__Topic 7 - Regular polygons __ /__ /__Topic 8 - Exterior angles of polygons __ /__ /__

Topic 14 - Similar triangles __ /__ /__

Topic 9 - Congruent triangles __ /__ /__

Topic 15 - Using similar triangles __ /__ /__

Topic 10 - Results from congruent triangles __ /__ /__

Topic 16 - Intercept properties of transversals __ /__ /__

Topic 11 - Special triangles __ /__ /__

Topic 17 - Pythagoras’ theorem __ /__ /__

Topic 12 - Properties of quadrilaterals __ /__ /__

Topic 18 - Area __ /__ /__

Topic 13 - Tests for special quadrilaterals __ /__ /__

Topic 19 - Solving geometrical problems __ /__ /__

Date completed

Author of The Topics and Topic Tests: AS Kalra

iiPlane geometry

Mathletics Instant Workbooks – Series L 2 Copyright © 3P Learning

1

Plane geometry

Plane geometryMathletics Instant Workbooks – Series L 2 Copyright © 3P Learning

Topic 1 - Notation, symbols, terms and definitions

40 EXCEL ESSENTIAL SKILLS: PRELIMINARY MATHEMATICS REVISION AND EXAM WORKBOOK

Plane geometry

QUESTION 2 Write the symbol that means:

a is congruent to ________ b is parallel to ________ c is similar to ________

d is perpendicular to ________ e therefore ________ f triangle XYZ ________

QUESTION 1 Name the marked angle.

a b c d

________________ ________________ _______________ ________________

Notations, symbols, terms and definitionsEXCEL PRELIMINARY MATHEMATICS

Ch. 2.1, p. 30

QUESTION 4 Write down the letter that appears beside the correct diagram.

A QP B QP C QP

a line PQ ______________ b interval PQ ____________ c ray PQ _______________

QUESTION 3 Complete each statement.

a A _______________________ angle measures 90° .

b _______________________ means at right angles.

c Bisect means ___________________________.

d _______________________ points are points on the same straight line.

e _______________________ lines are lines in the same plane that never meet.

f A scalene triangle has _______ equal sides and ______ equal angles.

g An isosceles triangle has _______ equal sides and ______ equal angles.

h An equilateral triangle has _______ equal sides and ______ equal angles.

i A quadrilateral is any _____ sided figure.

j A ________________________ is a quadrilateral with both pairs of opposite sides parallel.

k A rectangle is a parallelogram with one angle _______________________.

l A _______________________ is a parallelogram with one pair of adjacent sides equal.

n A _______________________ is a rectangle with two adjacent sides equal.

n A _______________________ is a quadrilateral with one pair of opposite sides parallel.

o A _______________________ is a many-sided plane figure.

p A _______________________ polygon has all its sides equal and all its angles equal.

q An _______________________ is a perpendicular line from the base of a figure to its highest point.

A

B C

F

E

D

P

S

R

Q

O K

N

M

L

CHAPTER 2

2

Plane geometry


Topic 2 - Angles (1)

41CHAPTER 2 – Plane geometry

Plane geometry

QUESTION 2 Find the value of the pronumeral.

a b c

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

d e f

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

g h i

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

j k l

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

QUESTION 1 Complete.

a A straight line measures ____________.

b Complementary angles add up to ____________.

c Supplementary angles add up to ____________.

Angles (1)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.2, 2.3, p. 30

x°68°

x°129°x°

51°

21°

x°37°

x°31°

19°x°

56°

A

O D

B C

x°27°Q

P S

T

Rx°

41°

158°a°

a°

e°e° e° 2x°3x°

2n°3n°4n°

3

Plane geometry




Plane geometry

QUESTION 1 Complete each statement.

a An acute angle measures ______________________________.

b An obtuse angle measures ______________________________.

c A reflex angle measures ______________________________.

d Two pairs of vertically opposite angles are formed when two _______________________________ intersect.

e Vertically opposite angles are ______________________________.

f A complete revolution measures ______________________________.

g Angles which meet at a point total ______________________________.


Ch. 2.4, 2.5, p. 31

QUESTION 2 AB and CD are straight lines. Find the value of x, giving reasons.

a b c

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

d e f

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

g h i

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

x°79°

A

BC

D

x°147°

x°156°

A

BC

D

x°125° x°

317°x°

128° 77°55°

x°84°

A

BC

D

39°FE

2x°6x°

4x°3x° (7x – 20)°(5x + 2)°

A

BC

D

4

Plane geometry




Plane geometry

QUESTION 1 Give reasons in each case.

a ABC is a straight line. BE bisects ∠FBD. Find the size of ∠ABE

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

b PS and UR are straight lines intersecting at O. Find ∠POQ

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

c AGD and BGE are straight lines. Determine whether FGC is a straight line.

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________


Ch. 2.2, 2.3, 2.4, 2.5, p. 30–31

72°A B

F

C

D

E

42°

32°

U

QP

R

S

T

27°O

59°

A

B

F

C

67°

D

G

E

54°

5

Plane geometry


Topic 3 - Corresponding, alternate and co-interior angles (1)


Plane geometry

Corresponding, alternateand co-interior angles (1)

EXCEL PRELIMINARY MATHEMATICS

Ch. 2.6, p. 32

QUESTION 2 Find the value of each pronumeral. Give reasons.

a b c

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

d e f

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

QUESTION 3 Determine whether AB is parallel to CD. Give reasons.

a b c

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

QUESTION 1 Choose the letter that names the type of angle shown.

A alternate B co-interior C complementary D corresponding

a b c d

________________ ________________ _______________ ________________

x°

129°A B

C D

EF

G

H

x°83°

M

K

J

P

I

ON

L

x°95°

SU

XR

QV

W

T

x° 66°

E

CA

H

D

G

B

F

x°71°

A

G

ED

HC

F

B

y°

z° x° 125°

A

GE

D

H

C

F

B

y°z°

125°BF

CG D

E

A

H135°

94°H

C

E G

D

F

A

B

86°69°

H

CE

G

D

F

A

B

69°

6

Plane geometry


Topic 3 - Corresponding, alternate and co-interior angles (2)


Plane geometry

Corresponding, alternateand co-interior angles (2)


Ch. 2.6, p. 32

QUESTION 1 Find the size of the required angles, giving reasons.

a AB || CD || EF∠FBC = 47°∠BCD = 117°Find ∠BFE

____________________________________________________________

____________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

b ∠EFB = 105°∠FBC = 28°Find ∠DCB

____________________________________________________________

____________________________________________________________

____________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

c ABCD is a straight lineEB || CJFG || HI∠ABE = 60°∠ACH = 100°Find ∠ICJ

____________________________________________________________

____________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

EA

F

D

B

C

B

E

C

H

A

J

F

D

IG

E

BA

C D

F

7

Plane geometry


Topic 4 - Angle sum of a triangle and the exterior angle of a triangle


Plane geometry

QUESTION 2 Give reasons.

a b

AB and CD intersect at E. Find ∠ECB Find ∠FHJ

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

A n g l e s u m o f a t r i a n g l eand the exterior angle of a triangle


Ch. 2.7, 2.8, p. 33

QUESTION 1 Find the value of x.

a b c

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

d e f

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

g h i

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

x°

39°43°

x°

69° 51°

x°

64°

x°

33°123°

x°2x°

87°

x°32°

88°

x° 40°

17°

x°

39°

112°

x°

125°

A

E

C

D

B

35°55°

F

G

J

H

I131°122°

8

Plane geometry


Topic 5 - Angle sum of a quadrilateral


Plane geometry

Angle sum of a quadrilateralEXCEL PRELIMINARY MATHEMATICS

Ch. 2.9, p. 34


a b c

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

d e f

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

g h i

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

QUESTION 2 The angles of a quadrilateral are in the ratio 1:2:3:4. Find the size of:

a the smallest angle b the other three angles

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

x°

110° 115°

70° x°

49° 68°

137° x°59°

142°

x°28°19°

100°

x°

123°

3x°

84°

76°

2x°

80°

70° x°

x°

40°280°

x°

160°45°x°

9

Plane geometry


Topic 6 - Angle sum of a polygon


Plane geometry

Angle sum of a polygonEXCEL PRELIMINARY MATHEMATICS

Ch. 2.10, p. 34

QUESTION 1 Find the angle sum of:

a a pentagon b an octagon c a dodecagon

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________

d a heptagon e a 16-sided polygon f a 20-sided polygon

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________

QUESTION 2 Find the number of sides of a polygon whose angle sum is:

a 1980° b 3060°

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

QUESTION 3 Find the value of x. Give reasons.

a b

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

x°78°

62° 47°

95°

H

I

J

K

G

x°

x° x°

x°

C D

E

FA

B

10

Plane geometry


Topic 7 - Regular polygons


Plane geometry

QUESTION 4 The diagram shows a square, four regular pentagons and four isosceles triangles.Find the size of ∠BNC

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

Regular polygonsEXCEL PRELIMINARY MATHEMATICS

Ch. 2.11, p. 35

I

K

D

F

A

E

C

G

J

M N

P O

BL

H

QUESTION 2 Find the size of each angle of a regular:

a hexagon b decagon c dodecagon

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________

d nonagon e 15-sided polygon f 30-sided polygon

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________

QUESTION 3 Find the number of sides of a regular polygon if each angle measures:

a 135° b 160°

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

QUESTION 1 A regular polygon is one in which all the sides are _________________________ and all the

angles are _________________________.

11

Plane geometry


Topic 8 - Exterior angles of polygons


Plane geometry

Exterior angles of polygonsEXCEL PRELIMINARY MATHEMATICS

Ch. 2.12, p. 35

QUESTION 5 Sides HA and CB of a regular octagon are produced to P as shown in the diagram.Find the size of ∠APB

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

QUESTION 3 Find the size of the exterior angle of a regular:

a hexagon b decagon c 15-sided polygon

________________________ ________________________ _________________________

________________________ ________________________ _________________________

________________________ ________________________ _________________________


a b

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

QUESTION 1 The sum of the exterior angles of any polygon is _________________________.

QUESTION 4 For a regular 20-sided polygon, find the size of each:

a exterior angle b interior angle

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

D

E

B

C

F

G

H

A

P

81°

72°

38°29°

84°

x°

70°50°

80°

x°

12

Plane geometry


Topic 9 - Congruent triangles (1)


Plane geometry


a Two figures are congruent if they are exactly the same _________________ and _________________.

b Two triangles are congruent if:

i ________________________________________________________________________________

________________________________________________________________________________ (SSS)

ii ________________________________________________________________________________

________________________________________________________________________________ (SAS)

iii ________________________________________________________________________________

________________________________________________________________________________ (AAS)

iv ________________________________________________________________________________

________________________________________________________________________________ (RHS)

Congruent triangles (1)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.13, p. 36

QUESTION 2 Determine whether the two triangles are necessarily congruent. If they are congruent, writedown the reason (SSS, SAS, AAS, RHS).

a b c

_______________________ _______________________ ________________________

d e f

_______________________ _______________________ ________________________

g h i

_______________________ _______________________ ________________________

50°

50°

60° 50°

9 cm60°

70°9 cm

13

Plane geometry


Topic 9 - Congruent triangles (2)


Plane geometry

Congruent triangles (2)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.13, p. 36

QUESTION 1 Complete these proofs.

a BE = AD b QS and PT intersect at R

In ∆s ABE, EDA In ∆s PQR, TSR

∠BAE = _______ _________ (both 90° , given) ∠PQR = _________ _______________________

BE = __________ ______________________ ∠QRP = __________ _______________________

AE = __________ ______________________ PR = _________________ ________________

∴ ∆ABE ≡ ∆EDA ________________ __________________________________________

QUESTION 2 Prove the given pair of triangles congruent.

a b

AB is parallel to DC PQRS is a rectangle. T is a point on PQ and U aAB = DC point on SR such that TQ = SUProve ∆ABC is congruent to ∆CDA Prove ∆PSU is congruent to ∆RQT

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

c d

AB = CD and AC = BD JKLM is a square and JP = JQProve ∆ABC is congruent to ∆DCB Prove ∆JKP is congruent to ∆JMQ

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

P

Q

T

S

R

A

B

C

D

E

D

A B

C S

P Q

R

T

U

A

B C

D M

J

P

L

K

Q

14

Plane geometry



Plane geometry

QUESTION 1 If two triangles are congruent then the corresponding sides are ________________ and the

corresponding angles are ________________.


a AB = CD and BC = DA

i Show that ∆ABC is congruent to ∆CDA

________________________________________________________

________________________________________________________

________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

ii Hence show that AB is parallel to DC

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

b AB is perpendicular to ADAD = DCBD bisects ∠ADCFind the size of ∠BCD

____________________________________________________________

____________________________________________________________

____________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

Results from congruent triangles (1)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.14, p. 36

D

A B

C

D

A B

C

Topic 10 - Results from congruent triangles (1)

15

Plane geometry



Plane geometry

Results from congruent triangles (2)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.14, p. 36


a JM is perpendicular to ML and KL is perpendicular to MLN is the midpoint of ML∠KJN = ∠JKN

i Show that JN = KN

________________________________________________________

______________________________________________________________________________________

ii Show that ∆JMN is congruent to ∆KLN

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

iii Find the size of ∠MJK

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

b PQRS is a trapezium (PQ || SR)T is the point of intersection of the diagonals∠QPT = ∠PQT

i Show that ST = RT

________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

ii Show that ∆PST is congruent to ∆QRT

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

iii Hence show that PS = QR

______________________________________________________________________________________

M

J

N

K

L

S

P Q

R

T

Topic 10 - Results from congruent triangles (2)

16

Plane geometry



Plane geometry


a An _______________________________ triangle has two equal sides and two equal angles. The angles

opposite the equal sides are _______________.

b An _______________________________ triangle has three equal sides and three equal angles. Each angle

measures _________.

Special trianglesEXCEL PRELIMINARY MATHEMATICS

Ch. 2.15, p. 37


a b c

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

d e f

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________

_______________________ _______________________ ________________________


a PQ = PR = QS b ∠BAC = ∠EBC = α∠PSQ = 62° ∠ABD = ∠ECB = βFind ∠SPR Show that CE = CD

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

x° x°

73°

x°

40°

x°

65°

x° x°68°

62°

P

RSQ

E

B

D CA

α

β

α

β

Topic 11 - Special triangles

17

Plane geometry



Plane geometry

Properties of quadrilaterals (1)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.16, p. 37

QUESTION 2 Find the value of each pronumeral.

a b

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

c d

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

e f

________________________________________ __________________________________________

________________________________________ __________________________________________

________________________________________ __________________________________________

QUESTION 1 Determine whether the following properties are true or false.

a The opposite sides of a parallelogram are equal. ________

b A square has all the properties of a rectangle. ________

c The opposite angles of a parallelogram are equal. ________

d The diagonals of a parallelogram bisect each other. ________

e A rhombus has all the properties of a parallelogram. ________

f The diagonals of a rhombus bisect each other at right angles. ________

g A parallelogram has all the properties of a rectangle. ________

h The diagonals of a rhombus bisect the angles through which they pass. ________

i A square has all the properties of a parallelogram. ________

j The diagonals of a rectangle are equal. ________

k A square has all the properties of a rhombus. ________

a°

73° b°

y cm

9 cm

x cm

5 cm

y°x°54° y m

15 m

17 m

x m8 cm

b m

4 m

7 m

a mx°

Topic 12 - Properties of quadrilaterals (1)

18

Plane geometry



Plane geometry

QUESTION 1

a Find the value of x

___________________________________________________________

___________________________________________________________

___________________________________________________________

b Find the length of the diagonal PR

___________________________________________________________

___________________________________________________________

___________________________________________________________

Properties of quadrilaterals (2)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.16, p. 37

R

Q

(9x – 17) cm

P

S

(5x + 23) cm

QUESTION 2


___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

b Find the length of the side BC

___________________________________________________________

___________________________________________________________

___________________________________________________________

c What type of quadrilateral is ABCD? Justify your answer.

_______________________________________________________________________________________

QUESTION 3


___________________________________________________________

___________________________________________________________

___________________________________________________________

b Find the size of ∠MJK

___________________________________________________________

___________________________________________________________

___________________________________________________________

c What type of quadrilateral is JKLM? Justify your answer.

_______________________________________________________________________________________

(7x – 43)°

M

K

L

J(4x + 14)°

61 m

(11x – 16) m

A B

CD

(8x + 5) m

Topic 12 - Properties of quadrilaterals (2)

19

Plane geometry



Plane geometry

QUESTION 3

a E is the point of intersection of the diagonals of ABCDAC ⊥ BD AE = EC BE = ED

i Which test is used to prove ∆ABE ≡ ∆CBE? ____________________________

ii Which test is used to prove ∆ABE ≡ ∆CDE? ____________________________

iii What conclusion can be drawn about sides AB, BC and CD?

_______________________________________________________________

iv What type of quadrilateral is ABCD?__________________________________

b PQRS is a parallelogramPR = QS

i Which test is used to prove ∆PSR ≡ ∆QRS? ___________________

ii Why does ∠PSR = ∠QRS?

______________________________________________________

iii Find the size of ∠PSR

______________________________________________________

______________________________________________________

iv What type of quadrilateral is PQRS? ________________________

Tests for special quadrilateralsEXCEL PRELIMINARY MATHEMATICS

Ch. 2.17, p. 38

QUESTION 2 What special type of quadrilateral is shown? Justify your answer.

a b c d

_________________ __________________ ________________ __________________

_________________ __________________ ________________ __________________

_________________ __________________ ________________ __________________

QUESTION 1 Choose the best answer from parallelogram, rhombus, rectangle or square.

a A quadrilateral with both pairs of opposite sides equal __________________________

b A quadrilateral with all its sides equal __________________________

c A parallelogram with both diagonals equal __________________________

d A quadrilateral with one pair of opposite sides equal and parallel __________________________

e A quadrilateral whose diagonals bisect each other __________________________

f A quadrilateral with both pairs of opposite angles equal __________________________

g A quadrilateral whose diagonals bisect each other at right angles __________________________

AB

CD

E

P Q

RS

Topic 13 - Tests for special quadrilaterals

20

Plane geometry



Plane geometry

QUESTION 2 State whether the two triangles are similar. Justify your answer.

a b c

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

d e f

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

QUESTION 3 Show that the triangles are similar.

a ∆ADE, ∆ACB b ∆PQT, ∆RQS

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

S i m i l a r t r i a n g l e sEXCEL PRELIMINARY MATHEMATICS

Ch. 2.18, p. 39

QUESTION 1 Similar figures are exactly the same ________________________ but not necessarily

the same ________________________.

3

56 6

12

10

29°61°

2

85

11

54°95°54° 41°

4

5

6

16

2536

D

A E B

C

Q

P

R

ST

20 m12 m

18 m30 m

6

6.5

12

13

Topic 14 - Similar triangles

21

Plane geometry



Plane geometry

U s i n g s i m i l a r t r i a n g l e s ( 1 )EXCEL PRELIMINARY MATHEMATICS

Ch. 2.18, p. 39

QUESTION 2 Given that the two triangles are similar, find the value of each pronumeral.

a b

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

c d

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

QUESTION 1 If two triangles are similar then the corresponding ________________________ are equal and

the corresponding ________________________ are in proportion.

a20

28

b

7

16

x 15

12

y

8

9

a

5

b18

2

x 56

63y

40 35

Topic 15 - Using similar triangles (1)

22

Plane geometry



Plane geometry

U s i n g s i m i l a r t r i a n g l e s ( 2 )EXCEL PRELIMINARY MATHEMATICS

Ch. 2.18, 2.19, p. 39–40

QUESTION 1 Given that the two triangles are similar, find the value of the pronumerals.

a b

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

QUESTION 2 Show that the triangles are similar and hence find the value of each pronumeral.

a b

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

xx + 518

2y

12

y + 4

a

5

8b

4

a + 3

x

10

15

1413

y

B

A

C

D

E

aQ

15

8

5bT

S

9

P

R

Topic 15 - Using similar triangles (2)

23

Plane geometry



Plane geometry

Intercept prope rties of transversals (1)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.20, p. 40


a b

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

c d

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________


a Parallel lines preserve the __________________________ of intercepts on transversals.

b If a family of parallel lines cuts equal intercepts on one transversal, it does so on all

__________________________.

x 4

9 6

a 3

125

x15

a

3628

b

35

42

Topic 16 - Intercept properties of transversals (1)

24

Plane geometry



Plane geometry

Intercept properties of transvers al s (2)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.20, p. 40


a b

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

QUESTION 2 AC = 35 cm DE = 16 cm EF = 14 cmFind the length of AB

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

5

x

5 4

9

y

x25

15y

20 20

C

B

A D

E

F

Topic 16 - Intercept properties of transversals (2)

25

Plane geometry



Plane geometry

QUESTION 2 Find the length of the hypotenuse of each triangle.

a b c d

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

Pythagoras’ theorem (1)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.21, p. 41

QUESTION 1 Write down the name of the hypotenuse of each triangle.

a b c d

_________________ __________________ _________________ _________________

QUESTION 3 Find the length of the unknown side.

a b c d

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

B C

A E

F

D I

G

H K

LJ

x m

5.6 m

3.3 mx km

40 km 399 km x cm5 cm

8 cmx m

96 m

57 m

QUESTION 4 Find the length of the side marked x.

a b c d

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

x m51 m

45 m

x m

3.9 m8.9 mx km

39 km13 km x cm8 cm

12 cm

x m19 m

180 m

x m149 m

140 m x km

29 km9 km

x mm

14 mm 37 mm

Topic 17 - Pythagoras’ theorem (1)

26

Plane geometry



Plane geometry

QUESTION 4 Find the perpendicular height of the triangular prism.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

Pythagoras’ theorem (2)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.21, p. 41

QUESTION 1 Each triangle is right-angled. True or false?

a b c d

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

QUESTION 2 Find the exact length, in simplest form, of the side marked x.

a b c

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

QUESTION 3 ABCDEFGH is a rectangular prism. Find the length of:

a AC b AH

________________________ ________________________

________________________ ________________________

________________________ ________________________

________________________ ________________________

________________________ ________________________

35 m

122 m

127 m

145 cm

24 cm143 cm

85 km

77 km

36 km46 m

55 m

23 m

x m7 m

5 mx m

6 m

4 m

x cm

11 cm

7 cm

9 cm

12 cm

8 cmA

F E

D

H

CB

G

714 mm

672 mm295 mm

Topic 17 - Pythagoras’ theorem (2)

27

Plane geometry



Plane geometry

Ar e a ( 1 )EXCEL PRELIMINARY MATHEMATICS

Ch. 2.22, p. 42

QUESTION 1 Find the area.

a b c d

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

e f g h

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

i j k l

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

_________________ __________________ _________________ _________________

QUESTION 2 Find the area of each composite shape.

a b c

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

8 m

5 m

15 m

5 m3 m

32 cm 15 m

7 m

14 m

5 m

55 mm

37 mm

15 m

12 m

8 m9 cm

7 cm

9 cm

8 cm17 cm10 cm

3 cm

8 cm 6 cm

5 km

10 km

13 km12 km 15 m12 m

20 m

20 m

11 m

7 m

19 mm

66 mm 34 mm

41 mm

13 km

4 km

Topic 18 - Area (1)

28

Plane geometry



Plane geometry

Area (2)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.22, p. 42

QUESTION 2 Find the area in hectares.

a b c

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

________________________ ________________________ ________________________

QUESTION 3 Find each area.

a b

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

QUESTION 1 Find the shaded area.

a b

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

_______________________________________ __________________________________________

3.6 m5.7 m

19.4 m

11.2 m7.1 m9.8 m

15.7 m

23.5 m

200 m

900 m

50 m

80 m

41 cm

40 cm

75 m42 m

65 m

125 m

120 m

Topic 18 - Area (2)

29

Plane geometry



Plane geometry

QUESTION 2 AB = AF. AB is produced to C and AF is produced to E.D is a point on CE such that FE = FD and BF bisects ∠AFD.Let ∠ABF = α

a Find the size of ∠DFE in terms of α

______________________________________________________________

______________________________________________________________

______________________________________________________________

b Show that AC is parallel to FD.

______________________________________________________________

______________________________________________________________

______________________________________________________________

c Find ∠FDE in terms of α

______________________________________________________________

______________________________________________________________

d What type of quadrilateral is BCDF?

______________________________________________________________

______________________________________________________________

______________________________________________________________

e Show that BC = FE

______________________________________________________________

______________________________________________________________

______________________________________________________________

Solving geometrical problems (1)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.23, p. 43

QUESTION 1 PQRS is a parallelogram. T is the midpoint of side PS. TQ = TR.

a Show that triangles PQT and SRT are congruent.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

b Find the size of angle TPQ.

______________________________________________________________

______________________________________________________________

______________________________________________________________

c What conclusion can be drawn about PQRS?

______________________________________________________________

P Q

T

S R

A CB

F D

E

Topic 19 - Solving geometrical problems (1)

30

Plane geometry



Plane geometry

Solving geometrical problems (2)EXCEL PRELIMINARY MATHEMATICS

Ch. 2.23, p. 43

QUESTION 1 AE and BD meet at C.Triangles ABC and CDE are right-angled at B and D respectively.ED = 3 m and AB = 9 m

a Show that triangles ABC and EDC are similar.

______________________________________________________________

______________________________________________________________

______________________________________________________________

b Show that AE = 4EC

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

c If AE = 20 m, find the length of BC

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

A

CB

D

E

Topic 19 - Solving geometrical problems (2)


Plane geometry

TOPIC TEST SECTION I

1 An angle measuring 212° would be

A supplementary B reflex C obtuse D acute

2 The two marked angles are

A alternate B co-interior

C corresponding D vertically opposite

3 The angle sum of a quadrilateral is

A 180° B 270° C 360° D 540°

4 In the diagram, x equals

A 20 B 80

C 100 D 140

5 Points that lie on the same line are

A collinear B corresponding C co-interior D concurrent

6 Two angles of a triangle measure 50° . The triangle must be

A scalene B isosceles C equilateral D right-angled

7 A polygon with seven sides is a

A heptagon B pentagon C dodecagon D hexagon

8 The exterior angles of a polygon add up to

A 90° B 180°

C 360° D depends on number of sides

9 Which test proves triangles ABC and EDC congruent?

A AAS B RHS

C SAS D SSS

10 A quadrilateral has diagonals which are not equal but which bisect each other at rightangles. The quadrilateral could be

A a square B a rectangle C a rhombus D any of these

Marks

1

1

1

1

1

1

1

1

1

Plane geoemetry

Instructions • This section consists of 15 multiple-choice questions

• Each question is worth 1 mark• Fill in only ONE CIRCLE for each question• Calculators may be used

Time allowed: 15 minutes Total marks = 15

1

60° 40°x°

A

CB

D

E

31Plane geometry


Plane geometryTopic Test PART AInstructions This part consists of 15 multiple-choice questions Each question is worth 1 mark Calculators may be used Fill in only ONE CIRCLE for each question

Time allowed: 15 minutes Total marks = 15


Plane geometry Marks

1

1

1

1


A 3 B 4

C 5 D 6

12 Which is correct? If two triangles are similar, then ...I the corresponding sides must be equalII the corresponding angles must be equal

A I only B II only C both I and II D neither I nor II

13 In the diagram, which is correct?

A a2 = b2 + c2 B b2 = a2 – c2

C c2 = a2 + b2 D a2 – b2 = c2

14 Which is not a property of parallelograms?

A opposite angles are equal B both pairs of opposite sides are parallel

C adjacent angles are supplementary D diagonals are equal

15 The area of the triangle is

A 60 m2 B 68 m2 C 127.5 m2

D there is insufficient information to find the area

Total marks achieved for SECTION I 15

1

3 9

x 12

a

b

c

8 m 15 m

17 m



1

1

1

1


A 3 B 4

C 5 D 6




A a2 = b2 + c2 B b2 = a2 – c2

C c2 = a2 + b2 D a2 – b2 = c2





A 60 m2 B 68 m2 C 127.5 m2



1

3 9

x 12

a

b

c

8 m 15 m

17 m

32Plane geometry


Plane geometryTopic Test PART A

15Total marks achieved for PART A

Plane geometryTopic Test PART B

33Plane geometry


Instructions This section consists of 25 questions Show all necessary working

Time allowed: 1 hour Total marks = 85


Plane geometry

30

16 Find the value of x.

a b c

___________________ ______________________ ___________________

d e f

___________________ ______________________ ___________________

g h i

___________________ ______________________ ___________________

j k l

___________________ ______________________ ___________________

m n o

___________________ ______________________ ___________________

17 Find the size of the required angle.

a AC is parallel to DG __________________________________

∠DEB = 155° __________________________________

∠BFG = 132° __________________________________

Find ∠EBF __________________________________

TOPIC TEST SECTION II

Plane geometry

Instructions • This section consists of 11 questions

• Show all necessary working

Marks


x°69° x° 128° x° 45°

x°

81°58°124°

x°

118°

x°

84°

29°

x°59°

x°

117° x°82°

x°

51°

76°x°

94°

106°

87°

x° 28°

x° 48°x°

66°

x°

24°

D

AB

E

C

FG

155° 132°


Plane geometry

30

16 Find the value of x.

a b c

___________________ ______________________ ___________________

d e f

___________________ ______________________ ___________________

g h i

___________________ ______________________ ___________________

j k l

___________________ ______________________ ___________________

m n o

___________________ ______________________ ___________________

17 Find the size of the required angle.

a AC is parallel to DG __________________________________

∠DEB = 155° __________________________________

∠BFG = 132° __________________________________

Find ∠EBF __________________________________

TOPIC TEST SECTION II

Plane geometry

Instructions • This section consists of 11 questions

• Show all necessary working

Marks


x°69° x° 128° x° 45°

x°

81°58°124°

x°

118°

x°

84°

29°

x°59°

x°

117° x°82°

x°

51°

76°x°

94°

106°

87°

x° 28°

x° 48°x°

66°

x°

24°

D

AB

E

C

FG

155° 132°


34Plane geometry




3

17 b PR is parallel to TS __________________________________

QR = QS __________________________________

∠TSR = 105° __________________________________

Find ∠TSQ __________________________________

c WXYZ is a parallelogram __________________________________

WX = XY __________________________________

∠WXZ = 24° __________________________________

Find ∠XWZ __________________________________

d ABCD is a parallelogram __________________________________

AB = AC __________________________________

∠BAC = 20° __________________________________

Find ∠ADC __________________________________

18 Find:

a the angle sum of a pentagon b the size of each angle of a regular pentagon

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

19 Determine whether AB is parallel to CD.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

20 Find the length of side BC.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________3

Z

W X

Y

24°

B

E

A

D

C

FG H

82°108°

B

A

C

24 m 51 m

6

12

R

QP

S

T

D

A B

C



3

17 b PR is parallel to TS __________________________________

QR = QS __________________________________

∠TSR = 105° __________________________________

Find ∠TSQ __________________________________

c WXYZ is a parallelogram __________________________________

WX = XY __________________________________

∠WXZ = 24° __________________________________

Find ∠XWZ __________________________________

d ABCD is a parallelogram __________________________________

AB = AC __________________________________

∠BAC = 20° __________________________________

Find ∠ADC __________________________________

18 Find:

a the angle sum of a pentagon b the size of each angle of a regular pentagon

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

19 Determine whether AB is parallel to CD.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

20 Find the length of side BC.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________3

Z

W X

Y

24°

B

E

A

D

C

FG H

82°108°

B

A

C

24 m 51 m

6

12

R

QP

S

T

D

A B

C



1

1

1

1


A 3 B 4

C 5 D 6




A a2 = b2 + c2 B b2 = a2 – c2

C c2 = a2 + b2 D a2 – b2 = c2





A 60 m2 B 68 m2 C 127.5 m2



1

3 9

x 12

a

b

c

8 m 15 m

17 m


35Plane geometry




21 Determine whether triangle PQR is right-angled.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

22 Triangles ABC and DEF are similar.

a Find x. b Find y.

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

23 ABCD is a rhombus.∠BAD = 114°

a Find the size of ∠BDC b If BD = 14 cm and AC = 9 cm,find the area of ABCD

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

24 PQRS is a rectangle. T is a point inside the rectangle such that PT = ST

a Show that ∠QPT = ∠RST

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

6

6

A B

CD

3

R

P

Q

72 mm 135 mm

153 mm

A

B

C

91 m

78 m

x m

D E

F

85 m119 m

y m

P Q

RS

T



21 Determine whether triangle PQR is right-angled.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

22 Triangles ABC and DEF are similar.

a Find x. b Find y.

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

23 ABCD is a rhombus.∠BAD = 114°

a Find the size of ∠BDC b If BD = 14 cm and AC = 9 cm,find the area of ABCD

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

24 PQRS is a rectangle. T is a point inside the rectangle such that PT = ST

a Show that ∠QPT = ∠RST

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

6

6

A B

CD

3

R

P

Q

72 mm 135 mm

153 mm

A

B

C

91 m

78 m

x m

D E

F

85 m119 m

y m

P Q

RS

T


36Plane geometry


85Total marks achieved for PART B



4

85Total marks achieved for SECTION II

24 b Prove that triangles PQT and SRT c Show that ∆TQR is isosceles.are congruent.

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

25 BE is perpendicular to AD. CD is perpendicular to AD.AE = 4 cm BE = 6 cm CD = 9 cm

a Show that triangles ABE and ACD are similar.

_______________________________________________________

_______________________________________________________

_______________________________________________________

b Find the length of ED c Find the exact length of AC

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

26 JKLM is a quadrilateral with JK parallel to ML.JK = 8 m LM = 15 m JM = KL = 9.1 mFind the area of JKLM.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

J

L

K

M

B

DEA

C

4

69

6

6



4

85Total marks achieved for SECTION II

24 b Prove that triangles PQT and SRT c Show that ∆TQR is isosceles.are congruent.

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

25 BE is perpendicular to AD. CD is perpendicular to AD.AE = 4 cm BE = 6 cm CD = 9 cm

a Show that triangles ABE and ACD are similar.

_______________________________________________________

_______________________________________________________

_______________________________________________________

b Find the length of ED c Find the exact length of AC

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

26 JKLM is a quadrilateral with JK parallel to ML.JK = 8 m LM = 15 m JM = KL = 9.1 mFind the area of JKLM.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

J

L

K

M

B

DEA

C

4

69

6

6

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