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---

Properties of Quadrilaterals

1

Interior angles add up to 360

2

All interior angles are right angles

3

All sides are equal

4

Both diagonals are equal

5

The diagonals are perpendicular

6

The diagonals bisect each other

7

Both diagonals bisect the angles they run
into

8

Only one diagonal bisects the other

9

Both pairs of opposite sides are equal

10

Both pairs of opposite sides are parallel

11

Exactly one pair of sides is parallel

12

Adjacent sides are equal

13

Each diagonal bisects the area of the
quadrilateral

14

The diagonals bisect each other
perpendicularly

15

Both pairs of opposite angles are equal

16

Exactly one pair of opposite angles is
equal

17

Exactly one pair of angles is bisected by
a diagonal

Trapezium

Kite

Square

Rectangle

PROPERTY

Rhombus

or a  in each box to indicate whether the quadrilateral named has the property described

Parallelogram

Put a

FAMILY OF QUADRILATERALS

Quadrilateral
Trapezium

Kite

Parallelogram

Rectangle

Rhombus

Square

Wynberg Boys’ High School
Department of Mathematics
Grade 10 – Revision of Grade 8 and 9 Geometry
1
...


Work out the value of x in the following
...


a)

b)

c)

x-24°

x

53°

3x
x

62°

3
...

AE = BE and AE
BCDE is a parallelogram
...


b)

Calculate the value of x
...


In each one of the following diagrams, find the size of the angle or side marked x or y:
a)

b)

C

A

36

32
E

B

x

D

y
x
B

A

D
20

c)

M
C

50

P

120
x

y

O

N

R

5
...

Find the value of x
...

SOME IMPORTANT TERMINOLOGY
1
...


Supplementary angles
Supplement

- two angles which add up to 180o
- the difference between 180o and a given angle

3
...


Complementary angles
Complement

- two angles which add up to 90o
- the difference between 90o and a given angle

5
...


Angles around a point add up to 360o
...


2
...


(  ’s on str
...


3
...


(vert
...
 ’s)

4
...

(Look for an “F” shape
...

(Look for a “Z” or “N” shape
...

(Look for a “C” or “U” shape
...
 ’s; ____ // ____)
Name the parallel lines
...

(co-int
...


Two lines, cut by a transversal, are parallel if…

5
...
two corresponding angles are equal
...
 ’s equal; ____  ____)
Name the equal angles
...


(alt  ’s equal; ____  ____)
Name the equal angles
...


(co-int
...
; ___ +___  180o )
Name the supplementary angles
...


The sum of the interior angles of a triangle is 180o
...


7
...


8
...


9
...


10
...


(ext
...

(____  ____; isosc
...

(opp
...

(  sum of quad
...


THE THEOREM OF PYTHAGORAS AND ITS CONVERSE AND COROLLARIES
RULE
Theorem of Pythagoras
In a right-angled triangle, the square on the hypotenuse is
equal to the sum of the squares on the other two sides
...


ABBREVIATED REASON

(Pythag
...


(converse of Pythag
...


Corollaries of the Theorem of Pythagoras
If the square on the longest side of a triangle is LESS
than the sum of the squares on the other two sides, then the
angle opposite the longest side is a ACUTE, and the
triangle is ACUTE-ANGLED
...


(corollary of Pythag
...


(corollary of Pythag
...


WYNBERG BOYS’ HIGH SCHOOL
GRADE 10 MATHEMATICS
REVISION (TEST PREP) WORKSHEET

GEOMETRY (INCLUDING PYTHAGORAS)
Note: This worksheet does not cover congruency and similarity
...
You must give reasons to justify
your statements
...

BOC
C

65o
A

(b)

ˆ  70o
In the diagram alongside, AC // DF, EBA
ˆ  2 x  30o
...

and BEF

x

D

O

P

A

70

B
o

C

2 x  30o
D

F

E
Q

(c)

ˆ  75o ,
In the given diagram, Pˆ  2x , Q

Q

ˆ  125o and PRS is a straight line
...

ˆ  135o and BOC
AOB
135o
A

O

x
C

(e)

The given diagram shows  ABC,

A

ˆ  x
...

AQP
o

105o

A

B

Q

PQRS is a straight line
...

and FG  FE
...


E

A
2

ˆ  2x and B
ˆ  4 x  10
...


4 x  10o
B

1 2

C

D

QUESTION 2
(a)

State the theorem of Pythagoras in words
...
]

(b)

ˆ  90o ,
In  PQR, shown alongside, Q

P

PQ  72 cm and QR  21 cm
...


(2)

Determine the perimeter of the triangle
...
To reach and work on high spots on the
construction site, the set-builders used a variety of cherry-picker type cranes, such as the one in the
picture
...
Note that CGFE
is a rectangle and that ACD is not a straight line
...
Show all necessary working and
give your final answer rounded off correctly to one decimal place
...

They record the measurements shown on the diagram below
...
YZ is the measurement taken along the ground
...

If it is not perpendicular, determine whether it is leaning towards the crane, or away from it,
giving a reason for your answer
...

As shown in the diagram, the circle touches each side of square ABCD, and the four corners of
PQRS lie on the circle
...

Determine the area of square PQRS
...
(Note that a maximum of two
marks will be awarded for the correct answer, if no working is shown
...
They have
finally decided that they want to replace the crooked fence with a single straight fence but
obviously they want the areas of each of their properties to remain the same
...

Drag B to D as shown
...
In each case your
diagram should be about 9 cm across
...
By the end of the exercise you will be required to cut out your quadrilateral and stick it in
the space provided to the left of the grid
...
Make whatever measurements or observations you need to enable you to decide whether the statements given are true or false for your
diagram, and record your results with ticks (not crosses) in the appropriate column
...
In some cases you will decide that the sentence could be true or false depending on
circumstances, so you would answer ‘maybe’
...

Record your name here:
Who else was in your group?

Your own case
PARALLELOGRAM

Paste your
parallelogram
here

Both pairs of opposite side are parallel
Both pairs of opposite sides are equal
Both pairs of opposite angles are equal
Just one pair of opposite angles is equal
All angles are right angles
Some, but not all, angles are right angles
The two diagonals are equal in length
The two diagonals bisect each other
Just one diagonal is bisected
The diagonals are perpendicular
Just one of the diagonals bisects the angles it runs into

Both diagonals bisect the angles they run into
Each diagonal is an axis of symmetry
Just one diagonal is an axis of symmetry

YES

The group decision
NO

YES

NO

Maybe

Your own case
TRAPEZIUM

Paste your
trapezium
here

YES

NO

The group decision
YES

NO

Maybe

Both pairs of opposite side are parallel
Both pairs of opposite sides are equal
Both pairs of opposite angles are equal
Just one pair of opposite angles is equal
All angles are right angles
Some, but not all, angles are right angles
The two diagonals are equal in length
The two diagonals bisect each other
Just one diagonal is bisected
The diagonals are perpendicular
Just one of the diagonals bisects the angles it runs into

Both diagonals bisect the angles they run into
Each diagonal is an axis of symmetry
Just one diagonal is an axis of symmetry
Your own case
RECTANGLE

Paste your
rectangle
here

Both pairs of opposite side are parallel
Both pairs of opposite sides are equal
Both pairs of opposite angles are equal
Just one pair of opposite angles is equal
All angles are right angles
The two diagonals are equal in length
The two diagonals bisect each other
Just one diagonal is bisected
The diagonals are perpendicular
Just one of the diagonals bisects the angles it runs into

Both diagonals bisect the angles they run into
Each diagonal is an axis of symmetry
Just one diagonal is an axis of symmetry

YES

NO

The group decision
YES

NO

Maybe

Your own case
RHOMBUS

Paste your
rhombus
here

YES

NO

The group decision
YES

NO

Maybe

Both pairs of opposite side are parallel
Both pairs of opposite sides are equal
Both pairs of opposite angles are equal
Just one pair of opposite angles is equal
All angles are right angles
Some, but not all, angles are right angles
The two diagonals are equal in length
The two diagonals bisect each other
Just one diagonal is bisected
The diagonals are perpendicular
Just one of the diagonals bisects the angles it runs into

Both diagonals bisect the angles they run into
Each diagonal is an axis of symmetry
Just one diagonal is an axis of symmetry
Your own case
SQUARE

Paste your
square
here

Both pairs of opposite side are parallel
Both pairs of opposite sides are equal
Both pairs of opposite angles are equal
Just one pair of opposite angles is equal
All angles are right angles
Some, but not all, angles are right angles
The two diagonals are equal in length
The two diagonals bisect each other
Just one diagonal is bisected
The diagonals are perpendicular
Just one of the diagonals bisects the angles it runs into

Both diagonals bisect the angles they run into
Each diagonal is an axis of symmetry
Just one diagonal is an axis of symmetry

YES

NO

The group decision
YES

NO

Maybe

Your own case
KITE

Paste your
kite
here

Both pairs of opposite side are parallel
Both pairs of opposite sides are equal
Both pairs of opposite angles are equal
Just one pair of opposite angles is equal
All angles are right angles
Some, but not all, angles are right angles
The two diagonals are equal in length
The two diagonals bisect each other
Just one diagonal is bisected
The diagonals are perpendicular
Just one of the diagonals bisects the angles it runs into

Both diagonals bisect the angles they run into
Each diagonal is an axis of symmetry
Just one diagonal is an axis of symmetry

YES

NO

The group decision
YES

NO

Maybe

QUADRILATERALS – Definitions
A quadrilateral is a plane, closed, four-sided figure
...
)
The sum of the interior angles is 360
...
For the purposes of this
exercise you can assume only what is explicitly given in the definitions here:

Parallelogram: the opposite sides are parallel
Trapezium:

just one pair of sides is parallel

Rectangle:

all the interior angles are 90

Rhombus:

all four sides are equal

Square:

all the interior angles are 90 and the four sides are all equal

Kite:

Two adjacent sides are equal to each other with one length, and the
other two adjacent sides are equal to each other but with a different
length
Use the coloured paper to make your figures:
Parallelogram
Trapezium
Rectangle
Rhombus
Square
Kite

=
=
=
=
=
=

White
Grey
Blue
Pink
Green
Brown

Study the rubric overleaf to see how you will be assessed
...

Alongside each name, including your own, put a ranking 1 to 4 according to the rubric given below
...

Level 4
PARTICIPATION
MANNER

CONTRIBUTION

Level 3

Worked hard with the
group at all times

Contributed well enough

Polite at all times, willing
to help those whose
understanding was weak,
enjoying the discussions

Contributed to
discussions in the
interests of the group as a
whole

A pivotal member of the
team, without whom the
task could not have been
done

An active participant
with useful contributions

Learner’s Name

Level 2

Level 1

Made some
contributions, but not
many
Only interested in
determining answers, but
not involved in proper
discussions
Provided some help or
insight, but generally
passive

Participation

Did not participate or
help at all
Dismissive of other boys’
problems or ideas, not
interested in the work
Made no meaningful
contribution to the task,
either through laziness or
because of a lack of
understanding

Manner

Contribution

Your own name goes in this block

TEACHER ASSESSMENT
Par’m Trapezium Rectangle Rhombus

Square

Kite

FIGURE: Correct shape/colour (1)
Appropriate size (1)
Care/neatness (1)
Evidence of investigation (1)
Individual GRID:
correspondence with pasted figure is
Poor/Satisfactory/Good (3)
Group GRID:
general accuracy is
Poor/Satisfactory/Good (3)

TOTAL:

SUMMARY
PEER

/12

CODE:
0-22

TEACHER

/60

23-29
30-59

TOTAL

/72

60-72

1 Not Achieved
2 Partially Achieved
3 Achieved
4 Excellent

Quadrilaterals and Isosceles Triangles
1
...
Find, giving
reasons clearly, the size of BTˆ C

A

B
T

F 125

C

D

2
...
Show all steps and give reasons for you answers
...


ABCD is a parallelogram, with BA
extended to P so that PA = PD
...


52
D

x

C

B

A

4
...
Calculate the value of x and of z
...


C

T

A

ABCD is a parallelogram
...


1

2

3

C

E
Figure 2

B

1

3

AE = BE and AEˆ B  52
...

a)

Calculate the value of y
...


A

B

y C

52°
x
E
7
...
in the following diagrams:
a)

B

b)

F

E
130°

D

a
c

35°

A

H

b

C

G
EFGH is a parallelogram

J

c)
L

P

60°

K

M

O

KLPM is a
parallelogram
...


d

N

d)
V
e

10°

R

S

70°

QRST is a rhombus
...


E

A

In the figure, AED and BFC are
straight lines
...

65
B

F

C

QUESTION 6
In the following diagram, AD  BC and AD // BC
...


C

B

(4)

b)

Hence prove that ABCD is a parallelogram
...

Aˆ  38  and CBˆ L  68 
...
Give a reason for each of your answers
...

Calculate, giving reasons,
the value of y
...


A

P

Q

The length of line segment AC is given by x and the
length of line segment BD is given by y
...
1
Using the formula for the area of a triangle,
show that the area of a kite can be given as:
Area (kite) = ½xy
(3)

D

y

B
M

x

5
...

(3)
5
...
4
Your Maths Teacher made a statement in class, “A kite that has equal diagonals
is a rhombus
...

(3)
[10]
1
...
1
...
1
...


(1)

ˆ D
ˆ
For quadrilateral ABCD it is given that B

1
...



If the apex angle is 15 larger than the base angles, find the size of each angle
...
In PQR ,
XY is parallel to QR,
PQ = 6 = PY,
2YR = 6
QR = 2PY
Note: PQR not drawn to scale

X

Q
a) Calculate: XQ ____________ (2)

Y

R

b) Calculate XY ____________

(2)

5
...
If false, give a reason for
your answer
...
1
...
1
...
1
...
1
...

All octagons are similar
...

All polygons are convex
...
1

In the sketch below, BA // QT and AP = AC
...
(8)

9
...
3
In the sketch below AD = AS = SR, AE // SM and RE // PM
...
3
...
3
...
1

Calculate the value of x and y, giving reasons:

W
x

S

10
24
T
25

y
2,8

R

V
(5)
7
...
3
reasons:
If AE

Prove that ΔABC is isosceles, giving
BC and E is the mid-point of BC
...
1
1
...
1

E

Find the values of the variables giving reasons:

C

(6)

50 

70 

b
c

(4)
1
...
2
A

B

50 

d
e

C

O

D

OB // DC
Ô is the centre of the circle
...
1
...
1
...
2

Are these lines parallel or not? Give reasons
...
2
...
2
...
2
...
3

The following pairs of triangles are similar, find the unknown sides
...

MNO  PQR
M
P
10

5

a

8

N

b

O

Q

7

R
[4]

1
...

Give reasons for your statements
...
5

M

N

P

Q

1
...
1 There are many ways to prove a quadrilateral is a parallelogram
...
both pars of opp
...

2
...
sides are equal and parallel
...
both pairs of opp
...

4
...

With this given information, explain why MNQP will form a parallelogram in the above sketch if
AM=MB, AN=NC, OQ=QC and OP=PB,
(4)
1
...
2

Explain with reasons why?
1
NO =
NB
...
1

Refer to Figure 1
...
1
...
1
...

The circle with centre O has a chord AB
...


Figure 3
A

2
...
2

Prove that ∆OAP  ∆OBP (4)
If AB = 8 cm and OP = 3 cm, determine the radius of the circle
...
1

Write down (without reasons) the size of the angles marked x and y in the following:
(a)
(b)

y

x y
32°

x
(4)

1
...
Reasons must be given
...
In each one of the following diagrams, find
the size of the angle or side marked x or y
(give reasons):
a)

A

b)

C

32

36
B

E

x

D

y

A

x
B

D

20

(4)

c)

(8)

M

50

P

120
x

y

(4)

O

N

[16]

R

QUESTION 2
...

Find the value of x (giving reasons)
...
From the given diagram:

C

a) prove that ABC is similar to DEC ;

2,5 cm

1,5 cm

D

b) calculate x, a and b, giving reasons;

E

a

c) prove that DEC  DEA
...


b

a)

B

name a six sided polygon
...


Find the size of an interior angle of a regular octagon
...


(3)
(3)

Decide whether or not the following are true:

a)
If two quadrilaterals have two equal opposite sides and two equal opposite angles, then it is a
parallelogram
...


2
...
What can be said about ABCD ?
EF 
2

Investigate parallelogram ABCD where E , F ,
G and H are the midpoints of the sides of the
parm
...
(no reasons necessary)
...
1

2 + 6 + 4 =
...
2

5=2+
...
3

1 + 7 + 9 + 11 =
...
4

3 + 5 + 8 + 10 + 12 =
...
5

1 + 3 + 5 + 7 + 8 =
...
1
Prove the theorem that states that the internal angles of a triangle
are supplementary
...
2

D

Find the value of x in the diagram below if AB=AC and BC=BE
...
1
List three properties of a rhombus that are not properties of a
parallelogram
...
2
PQRS is a rhombus with PS = PR = x mm
...
No reasons are required
...
2
7
...
2
7
...
3


...
3

R

(5)

In the figure, KMQR is a rhombus
...
Prove that MPˆ K  3MKˆ P

K

M

P
(6)
QUESTION 8:

Q

R

8
...


(5)

8
...


(5)

8
...


A

D
2x

x

8
...
1

Prove that ABCD is a parallelogram
...
3
...


(2)

8
...
3

Calculate x if AEˆ D  105



(4)

QUESTION 2:
2
...


(4)

2
...

Prove that DF  FC
B

A

D

C

F

(7)
QUESTION 3:

E

3
...


(6)

3
...


3
...
FO  JK and OG  KH
...
3
...


(5)

3
...
2

Now show that JOHK is a rectangle
...
1
Using the diagram below, prove the theorem that the diagonals of
a parallelogram bisect each other
...
2

P

(6)

In the diagram RSTV is a parallelogram
Y

R

S

O

V

W

T

4
...
1

Prove that ROY  TOW
...
2
...


(4)

MATHEMATICS CYCLE TEST
GRADE 10
Date: April 2006

Time: 60 min
Total: 70

Start question 1 and question 2 on separate exam pad sheets
...
3

Find the values of the variables giving reasons:

1
...
1
50 

70 

b
c

(4)
1
...
2
A

B

50 

d
e

C

O

D

OB // DC
Ô is the centre of the circle
...
1
...
1
...
4

Are these lines parallel or not? Give reasons
...
2
...
2
...
2
...
3

The following pairs of triangles are similar, find the unknown sides
...


MNO  PQR
M
P
10

5

a

8

N

b

O

Q

7

R
[4]

1
...

Give reasons for your statements
...
5

M

N

P

Q

1
...
3 There are many ways to prove a quadrilateral is a parallelogram
...
both pars of opp
...

2
...
sides are equal and parallel
...
both pairs of opp
...

4
...

With this given information, explain why MNQP will form a parallelogram in the above sketch if
AM=MB, AN=NC, OQ=QC and OP=PB,
(4)
1
...
4

Explain with reasons why?
1
NO =
NB
...
1
...


(3)

2
...
2

Find the gradient from A to B
...
1
...


(3)

2
...
4

Write down the equation of the straight line AB
...
1
...
Show all working
...
2

Monique works in the post office and can stamp 750 letters every 5 minutes
...
2
...


(2)

2
...
2 Work out and show in a table what amount of letters she will stamp
in 2 min, 3 min, 7 min, 8 min
...
2
...

(4)
[8]
2
...
3
...


(2)

2
...
2 m is positive and c is negative
...
)
[4]

2
...
Remember that in a parallelogram, diagonals bisect
...
4
...


(2)

2
...
2 Find S, one of the vertices of the parallelogram
...
5
Mr Boon is currently driving a Toyota Prado which cost him R690 000
new, in January 2004
...
By the end of 2006 the value of the
car is R640 000
...
The price of a Verso was R240 000 in the beginning
of 2006 and will cost R270 000 at the beginning of 2007
...
5
...
(2)
2
...
2 Mr
...
Find the point at which Mr Boon can sell his Prado and buy a brand new Verso without
having any money left
...
04
...
Identify the following polygons, stating:
a)
the name
b) whether it is convex or concave
i)

ii)

YIELD

iii)

iv)

(Use one of the white polygons on the ball)
(8)
2a)

H

THINK is a regular pentagon
...


1 2

1

K

x

2

N

F

(4)

b)
X

T

A

B

EXTEND is a regular
...
AN // BP
...


E

E

1 2
D

N

a
P

c) How many sides does a polygon have if the sum of its angles
is 2700?

(3)

(3)

3a)
A

What is wrong with this picture?
Explain
...
If false, explain why
...
(2)
ii) In ABC, Aˆ  90, AB = 3, BC = 4 and AC = 5
...

i) Write down the converse of this statement
...


(1)
(2)

4
...
AC  CD and CH  HD
...


5a)
L

ˆ  Lˆ ; G
ˆ A
ˆ
...

G
N
Is SG = TA?
Show all reasoning
...
Calculate AB
...


9,45 cm

10,5
cm

Above is a picture of the Arc de Triomphe in Paris
...
If the Arc
de Triomphe is 50m high, how wide is it? Show all working
...


P

S

ˆ D
ˆ
...

PQ // ST
...

R
S
P
T

Q

(4)

QUESTION 4

ABCD is a rectangle
...


B

A
Prove that :
OAB + OCD = OAD + OBC

O
D

C

QUESTION 1
Complete the following statements:
1
...
2

A rhombus is …

(2)

1
...
4

RSTU is a quadrilateral with  R = 4x,  S = 5x,  T = x and  U = 2x
...
4
...


1
...
2

Prove that RSTU is a trapezium
...

EC bisects  C
...


E

o

B

q
o

68

p
[5]
C

D

QUESTION 3
3
...


3
...
T is the midpoint of PR
...

STQ is a straight line
...
1
Prove the theorem that states that the
opposite angles of a parallelogram
are equal
...


X

4
...

H
Prove that DI = FH
...

Prove that  AXD is a right angle
...


Find a , b, c and d , giving reasons :
A

M

L

B

1
...
2
a
E
C

Q
130

20

D

O

N

P

(2)

(5)
/7/

QUESTION 2
2
...

A

2
...
2

I

B

x
x
x

G

x

3x  20
C

H

(3)

(5)

x

/8/

E

QUESTION 3

C
x

3

Find, giving reasons :
3
...


2 angles each equal to x

(2)
D

3
...
3

(2)

Hence calculate the value of x
...


/4/

C

D

QUESTION 5
5
5
...
2

Prove BCD = EDC

(4)
/7/

Grade 10

GEOMETRY REVISION EXERCISE

March 1998

NOTE: Draw diagrams on the right hand side of the page - it gives more writing space
...

Support all statements with reasons - using the accepted abbreviations
Present your argument succinctly (in a brief, concise, logical manner)
Parallel Lines
...

An isosceles triangle has two equal sides, and the angles opposite these sides are also equal
...
 DAB = 120o ,
DBC = 85o,  DCB = 65o
Prove that  ADB is isosceles
...

Prove that  P =  R
...

Prove that RT bisects  PRS
...


S
A

D
B

y

x
C
A

12

AB = BC
...
 BAD = 30o
and  ABD = x
...


30o

x
B
13

D

 SPR =  SRP = x
 QRP = 90o
Prove that S is the midpoint
of PQ
...
 ABC =  BDC = 90o
...


R

A
E
D

B

C

Congruency
Two triangles are congruent if they are identical in shape and size
...
Tested using the “ test”)
Two triangles can be proven to be congruent by proving one of the following sets of conditions
...

1
S
...
S
2
S
...
S
3
S
...
H
...

Approach:

Always state the triangles in which you are working - labelled in corresponding order
...

State that the triangles are congruent (  ) and add the condition for congruency used
...

eg

Prove that AB = CD given AB // CD
and BO = OC
...
)

C

D

B
A

15

O is the centre of the circle
...


O
C

D
16

AD = BC and  EDA =  FCB
Prove that  ACD =  BDC
...

Prove that  QPO =  RPO
...
2

Write down the letters a to e and then match the quadrilateral to the property:

PROPERTY
QUADRILATERAL
(a)
two pairs of adjacent sides equal
1
kite
(b)
four axes of symmetry
2
parallelogram
(c)
only one pair of sides parallel
3
rhombus
(d)
diagonals equal in length, but sides not
4
rectangle
necessarily equal in length
(e)
diagonals cross at 90o and bisect eadh other 5
square
6
trapezium

5

Find the value of x and y in the following figures
...

A

B

C

D

5
...
2

A
65 o

25 o
D

5
...
4

(4)

P

D

(4)

A
40 o

70 o E

x
C
6

D

Write down the 4 reasons for congruency
...
1

Write down the definition of a parallelogram
...
2
For each of 1
...
1 to 1
...
3 below, write down one property of a parallelogram
which has to do with its
...
2
...
2
...
2
...
3
Sketch a rhombus on your answer sheet
...
(Think of sides, angles, diagonals,
lines of symmetry, etc
...
Reasons need not be given in this
question
...
1

2
...
1
...

(1)
2
...
2 Write down the sizes of angles a and b
...
2
...

(1)
2
...
2 Write down the sizes of angles x and y
...


A

3
...

DEFB is a parallelogram
...


D

(6)

E

B

C

F
3
...


P
6x  12
(4)

2x + 40

3
...
Prove
S that
C is the midpoint of DE
...
)

(4)

D

C

E

Q

Do not re-draw the diagrams - Use the test question diagrams and hand them in stapled to the back of
your answers
...
There are 8 questions
...


1

F

a

H
G
L 30o M

[4]

T
2

TR = TS
...

Find the size of  b
...

Find the sizes of  c,  d and  e
...

BT bisects  ABC
...

Prove, with reasons, that PT = PB
...
T
...


X
 XYZ is drawn together with four other triangles
...
Lengths are in cm
...

Prove, using congruency, that
 ABC   DBC and that
 A =  D
...

PS = PQ
...


Q

T

x

=
x

S

R

[5]

1
...

Calculate the sizes of the
angles of the parallelogram,
giving reasons for all statements
...


K

In parallelogram KLMN
 K = x+60° and
 M = 3x-30°
...

N

M
(6)

3
...

Find the sizes of the angles
of DEFG
...


A

1999

B
O

C

D
A

2

B

Prove that AB = CD and AB // CD
...

Prove that AC = CB
...

Prove that CE = BD
...


A

/
x

=

=

B

7

Prove that AB = DC
and AB // DC
...


P

1
...

1
...
2

Q

Prove that PQRS is a rhombus
...


S

R
A

2
...

Calculate the angles of the parallelogram
DEFB
...


K

Calculate the angles of the parm KLMN,
if  N = 5x – 12° and  L = 3x + 18°

N

L

M

A

4
...

Calculate the sizes of
i)  D
ii)  A
iii)  AGK iv)  EKH

E

G

D

K

B

5
...
If FC = DC,
prove that ABCF is a parallelogram
...


ABDE is a parm
...


A

F

Prove: 1
...
FG = GC

2
...


B

C
F

3
...


E

A

D

C
B
A

4
...


Prove 1
...
BCEF is a parm
3
...


F

D

AGEF and ABCD are parms
...

Complete the following sentences, writing the words needed to complete the sentence
P
grammatically in the space provided on the answer sheet:
1
...

V
A rhombus is a ……
...
sides equal
T
A trapezium is a quadrilateral with …………
If one pair of opposite sides of a quadrilateral are equal
x 60°
and parallel then the quadrilateral is a ………………
x
A rectangle is a ……… with interior angles equal to ………
...
2
1
...
4
1
...
6
………

The diagonals of a rhombus ………… each other and are ……………
(10)

1
...


From the sketch alongside, name
(giving the vertices in alphabetical order)
2
...
2
2
...


one rhombus
three parallelograms
two trapezia (6)

For each of the sketches find the size of the angles marked with a small
letter
...

C

3
...
2
D

B

A

F

E

(7)

h
A

4
...
58°
In the figure,

D

22°

ˆ P
...
1

ˆR
the size of SV

4
...
3

the length of SQ

B

c

X

80°
d
C

(6)

U

k
m

V
122°

31°
E

W

5
...
Show your working
...
1

5
...
2

5
...

For each of the following figures, make an equation involving x, giving reasons, and then solve it
to find the value of x
...

State whether the following sentences are TRUE or FALSE
...

1
...
2
1
...
4
1
...
1
Give four properties of a quadrilateral such that each one on its own would guarantee
that the quadrilateral is a parallelogram
...
2

What extra properties do you need to prove that a parallelogram is a square?

(2)

2
...


ABCD is a square, with BFˆ D = 125
...


In the figure, AED and BFC are
straight lines
...

65
B

C

F

(6)

X

5
...

DB is produced to X and YC is drawn parallel to
AX to meet BD produced at Y
...
1

Prove that  AXO   CYO

(4)

5
...


(3)

Question 1
...
1

In a trapezium both pairs of opposite sides are parallel
...
2

The diagonals of a kite bisect each other at 900
...
3

If a parallelogram has equal diagonals then it must be a square
...
4

A square is always a rhombus, but a rhombus is only sometimes a square
...
5

If the diagonals of a quadrilateral are equal then it must be a rectangle
...


STRV and PURV are parallelograms
...


[6]
Question 4
...
1
Write down the ratio of the areas of the figures specified
...


ΔBDE : ΔABC

ΔAEC : parmABCD

A

4·1·1

4·1·2

E
D

A
D
E
B

C

B

C

(4)

4
...

Calculate:

B

A
G
E
D

F

C

4
...
1 the length of EF
4
...
2 the area of ΔGDC

4
...


Prove that Area ΔBCP  Area ΔABQ

(5)

D

A

B

P

C

Q

1
...
For those which are
FALSE, give a corrected version
...
1
1
...
3
1
...
5

The sum of all interior angles of a quadrilateral is 360
A trapezium has all sides of equal length
A kite has opposite sides parallel
The diagonals of a rectangle bisect the angles into which they run
If the diagonals of a quadrilateral cross at 90, the quadrilateral must be a square (9)

2
...

(4)
2
...
3

Name two quadrilaterals whose diagonals are perpendicular

(2)

B

A

1
...

Prove that it must be a rhombus

M

39

D

C

B

A

2
...

Prove that it must be a rhombus
...


C

B

A

In the figure ABCD is a parallelogram
...

ABCD is a rhombus, and DC is extended to P so
that BP = BC
...
1

DAˆ C

4
...


ABCD is a parallelogram
...
1

 ADM   CBP

5
...
3

MAˆ B  PCˆ D

M
D

C

Q

A
8
...

Prove

9
...
1

 ADP   CBQ

6
...

DP = BQ
...


P
D

C

B

A
P
10
...


Q

Prove that BPDQ is a parallelogram
...


ABCD is a parallelogram, and
DP  QB
...


P

Q

B

A
12
...


Q
V

Prove that APCQ is a parallelogram
...

ABCD is a parallelogram
...


Q

P
V
D

C

QUESTION 4

[3]

Refer to the diagram below, and solve for x (you must show all reasons)
...

5
...


(3)

5
...


(2)

5
...


(2)

5
...
Then prove your
conjecture
...


If a quadrilateral is a rectangle, it is not good enough to classify it as a parallelogram
...

7
...
2)

QUESTION 8

(2)

[11]

In the diagram below AD=AG and EF||DG
8
...


(3)

8
...


(2)

8
...


(3)

8
...
5)

Name 1 pair of similar triangles which are not congruent (do not prove they are similar) (1)

QUESTION 9
[2]
Do not attempt this
completed the other 8 questions
...

Prove that

AE AC

FB CD

question until you have



---