Download Properties and defintions File - Wynberg Moodle

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.
Write down (without reasons) the size of the angles marked x and y in the following:
a)
b)
y
x y
32°
x
2.
Work out the value of x in the following. Reasons must be given.
a)
b)
c)
x-24°
x
53°
3x
x
62°
3.
Refer to the figure alongside
A
B
y C
AE = BE and AÊB  52 .
BCDE is a parallelogram.
52°
x
Giving reasons:
E
a)
Calculate the value of y.
b)
Calculate the value of x.
D
4.
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.
In the figure, PQRS is a parallelogram with diagonals PR and SQ intersecting at O.
Find the value of x.
P
Q
6x
4x
80
S
O
R
WYNBERG BOYS’ HIGH SCHOOL
GRADE 10 MATHEMATICS
GEOMETRY REVISION:
GRADE 8 TERMINOLOGY AND RULES
Note: This information sheet does not cover congruency and similarity.
SOME IMPORTANT TERMINOLOGY
1.
2.
Supplementary angles
Supplement
- two angles which add up to 180o
- the difference between 180o and a given angle
3.
4.
Complementary angles
Complement
- two angles which add up to 90o
- the difference between 90o and a given angle
5.
Adjacent angles
- two angles which share a common arm, and have the same
vertex
SOME “RULES” OF GEOMETRY
RULE
ABBREVIATED REASON
1.
Angles around a point add up to 360o.
(  ’s around point ____)
Name the point.
2.
Angles on a straight line add up to 180o.
(  ’s on str. line ____)
Name the line.
3.
Vertically opposite angles are equal.
(vert. opp.  ’s)
4.
When a transversal cuts a pair of parallel lines, then…
(a)… pairs of corresponding angles are equal.
(Look for an “F” shape.)
AND
(b)… pairs of alternate angles are equal.
(Look for a “Z” or “N” shape.)
AND
(c)… pairs of co-interior angles are supplementary.
(Look for a “C” or “U” shape.)
(corres.  ’s; ____ // ____)
Name the parallel lines.
(alt  ’s; ____ // ____)
Name the parallel lines.
(co-int.  ’s; ____ // ____)
Name the parallel lines.
Two lines, cut by a transversal, are parallel if…
5.
(a)... two corresponding angles are equal.
(corres.  ’s equal; ____  ____)
Name the equal angles.
(b)…two alternate angles are equal.
(alt  ’s equal; ____  ____)
Name the equal angles.
OR
OR
(c)…two co-interior angles are supplementary.
(co-int.  ’s suppl.; ___ +___  180o )
Name the supplementary angles.
RULE
ABBREVIATED REASON
6.
The sum of the interior angles of a triangle is 180o.
(  sum of  ____)
Name the triangle.
7.
The exterior angle of a triangle is equal to the sum
of the two interior opposite angles.
8.
In an isosceles triangle the angles opposite the equal
sides are equal to each other.
9.
In a triangle, sides that lie opposite equal angles are
equal in length.
10.
In a quadrilateral (4-sided figure), the sum of the
interior angles is 360o.
(ext.  of  ____)
Name the triangle.
(____  ____; isosc.  ____)
Name the equal sides.
(opp. equal  ’s; ____  ____)
Name the equal angles.
(  sum of quad. ____)
Name the quadrilateral.
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.
Converse of the Theorem of Pythagoras
If the square on the longest side of a triangle EQUALS
the sum of the squares on the other two sides, then the
angle opposite the longest side is a RIGHT ANGLE.
ABBREVIATED REASON
(Pythag. in  ____)
Name the triangle.
(converse of Pythag. in  ____)
Name the triangle.
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.
If the square on the longest side of a triangle is GREATER
than the sum of the squares on the other two sides, then the
angle opposite the longest side is OBTUSE, and the
triangle is OBTUSE-ANGLED.
(corollary of Pythag. in  ____)
Name the triangle.
(corollary of Pythag. in  ____)
Name the triangle.
WYNBERG BOYS’ HIGH SCHOOL
GRADE 10 MATHEMATICS
REVISION (TEST PREP) WORKSHEET
GEOMETRY (INCLUDING PYTHAGORAS)
Note: This worksheet does not cover congruency and similarity.
NAME & SURNAME: ______________________________________________ CLASS: ________
QUESTION 1
Determine the value of x in each of the following diagrams. You must give reasons to justify
your statements.
(a)
ˆ  x , AOB
ˆ  90o ,
In the given diagram, COD
B
ˆ  65o and AOD is a straight line.
BOC
C
65o
A
(b)
ˆ  70o
In the diagram alongside, AC // DF, EBA
ˆ  2 x  30o . PBEQ is a straight line.
and BEF
x
D
O
P
A
70
B
o
C
2 x  30o
D
F
E
Q
(c)
In the given diagram, P̂  2x , Q̂  75o ,
Q
ˆ  125o and PRS is a straight line.
QRS
75o
125o
2x
P
(d)
S
R
In the diagram alongside, AOC is a straight line,
B
ˆ  x.
ˆ  135o and BOC
AOB
135o
A
O
x
C
(e)
The given diagram shows  ABC,
A
with   100o , B̂  45o , and Ĉ  x .
100o
45o
x
C
B
(f)
In the diagram alongside, AB // CD,
P
ˆ  105 and PRD
ˆ x.
AQP
o
105o
A
B
Q
PQRS is a straight line.
x
C
D
R
S
(g)
In the given diagram AE // BD, BA // CF
F
and FG  FE. F̂  30 and B̂  5x  20 .
o
o
30o
A
G 3
1 E
2
1 2
5 x  20o
B
(h)
1 2
D
C
In the diagram alongside AE // BD,
BA // CE and EC  ED.
E
A
2
ˆ  2x and B̂  4 x  10 .
CED
1
2x
o
Determine the value of x .
4 x  10o
B
1 2
C
D
QUESTION 2
(a)
State the theorem of Pythagoras in words. [HINT: In a right-angled triangle ...]
(b)
In  PQR, shown alongside, Q̂  90o ,
P
PQ  72 cm and QR  21 cm.
72 cm
Q
(1)
Determine the length of PR.
(2)
Determine the perimeter of the triangle.
21 cm
R
QUESTION 3
The diagram below shows a film set being constructed. 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.
(a)
In the diagram, lines have been drawn to indicate what measurements are known. Note that CGFE
is a rectangle and that ACD is not a straight line.
A
A
4m
G
4m
B
B
3m
3m
C
G
C
5m
5m
F
2m
E
3m
D
F
2m
E
3m
D
Using your knowledge of the theorem of Pythagoras, calculate the height (AF) at which a person
would be working when standing on the platform of the crane. Show all necessary working and
give your final answer rounded off correctly to one decimal place.
(b)
The set builders are concerned that the walls of the building are not perpendicular to the ground.
They record the measurements shown on the diagram below. XY is the measurement taken on the
wall of the building. YZ is the measurement taken along the ground.
X
X
4,5 m
4,5 m
4,0 m
4,0 m
Y
Y
Z
Z
2,5 m
2,5 m
Determine (by calculation) whether the wall of the building is perpendicular to the ground or not.
If it is not perpendicular, determine whether it is leaning towards the crane, or away from it,
giving a reason for your answer.
QUESTION 4
Consider the diagram given below, which shows a circle, with centre O, and two squares.
As shown in the diagram, the circle touches each side of square ABCD, and the four corners of
PQRS lie on the circle. The area of square ABCD is 256 cm2.
Determine the area of square PQRS. Show all your working. (Note that a maximum of two
marks will be awarded for the correct answer, if no working is shown.)
A
B
P
Q
O
S
D
R
C
A Fair Fence?
The Afrika family and Benjamin family have been neighbours for years and for as long as
they can remember they have had a crooked fence between their properties. 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.
Benjamin
Afrika
Can you help them by choosing where to build one straight fence (roughly where the
present fence is), but you must be able to show that the area of each property will stay the
same?
Fair Fence - SOLUTION
Join AC to form a triangle
Draw DE parallel to AC through B
You now have a triangle between parallel lines and its area remains the same as long as its
base remains the same.
Drag B to D as shown.
D
A
B
E
C
So triangle ADC has the same area as triangle ABC
Hence the one straight fence must be built along DC
D
A
C
E
QUADRILATERALS
You need: ruler, pencil, eraser, protractor, scissors, glue
Considering the definitions you have been given, and using your ruler, protractor etc, draw one example of each kind of quadrilateral on the coloured paper, as listed. In each case your
diagram should be about 9 cm across. You must also draw in the diagonals of each quadrilateral. 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. One of the reason for cutting it out is so that you can experiment with folding (HINT!), but you will probably find it easier to make
measurements from the figure before it is cut out. 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.
Parallelogram
=
White
Rectangle
=
Blue
Trapezium
=
Grey
Rhombus
=
Pink
Square
=
Green
Kite
=
Brown
When you have finished with your own shapes, you will get together with a group of three other boys and share your experience, and you will discuss as a group what you found in your
four individual cases; thus you will arrive at a group decision about which sentences must be true. In some cases you will decide that the sentence could be true or false depending on
circumstances, so you would answer ‘maybe’. Remember that we are looking for logical certainty here: if you say ‘YES’ then you must be convinced that the property you are talking about
is unavoidable, while ‘NO’ means it is impossible.
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.
(‘Plane’ means it can be drawn on a flat sheet of paper, ‘closed’ means that if the sides were fences, sheep could not get out.)
The sum of the interior angles is 360.
Some quadrilaterals have special features, and have special names. 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.
Note particularly what you will have to comment on about
the people who work with you in the group session – which
is also what they will be commenting on about you!
RUBRIC for QUADRILATERALS
PEER ASSESSMENT
In the grid below, list the boys who were in your group, putting your own name at the top of the list.
Alongside each name, including your own, put a ranking 1 to 4 according to the rubric given below.
You may be asked by your teacher to give some verbal justification for the codes you record.
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.
ABCD is a square, with BF̂D = 125. Find, giving
reasons clearly, the size of BT̂C
A
B
T
F 125
C
D
2.
Determine the size of the angles marked x. Show all steps and give reasons for you answers.
2·1
2·2
P
Q
B
A
x
42°
x
85°
S
(4)
R
D
M
2·3
N
E
C
(5)
AD=DE=EC
3x
O
L
x+10°
(5)
K
LM=MN
3.
ABCD is a parallelogram, with BA
extended to P so that PA = PD.
A
P
B
Calculate the value of x.
52
D
x
C
B
A
4.
x
ABCD is a parallelogram, and TB bisects
ABˆ C . Calculate the value of x and of z.
72
D
5.
C
T
A
ABCD is a parallelogram.
AB = AE and DC = EC
DAˆ E  40
z
D
40 
1
Calculate, with reasons, the sizes of:
1
6.
a)
Ê1
b)
Â1
c)
d)
Ĉ
Ê3
AE = BE and AÊB  52 .
BCDE is a parallelogram.
Calculate the value of y.
b)
Calculate the value of x.
3
C
E
Figure 2
B
a)
2
A
B
y C
52°
x
E
7.
D
Find the value of a, b, etc. 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.
KLMN is a kite.
d
N
d)
V
e
10°
R
S
70°
QRST is a rhombus.
QRVT is a kite
Q
e)
T
X
f)
Y
3f
C
2g+50°
A
W
f+10°
Z
B
3g+20°
7g+30°
D
E
8.
E
A
In the figure, AED and BFC are
straight lines.
D
115
65
Prove that ABCD is a parallelogram.
65
B
F
C
QUESTION 6
In the following diagram, AD  BC and AD // BC .
a)
D
A
Prove that ABC  CDA .
C
B
(4)
b)
Hence prove that ABCD is a parallelogram.
Page 3 of 15
QUESTION 2
A
a)
In the sketch alongside,
AC // KL and DC  BC .
Aˆ  38  and CBˆ L  68  .
C
1
38
2
2
D
K
1
1
2
68
B
Determine each of the following angles. Give a reason for each of your answers.
1)
B̂1
_________________________________________________
L
2)
B̂ 2
_________________________________________________
3)
Ĉ 2
_________________________________________________
4)
Ĉ1
_________________________________________________
b)
In the diagram alongside,
HONEYB is a regular hexagon
and BACH is a rhombus.
Calculate, giving reasons,
the value of y .
N
(4)
O
H
E
C
y
Y
A
B
QUESTION FIVE
A kite ABCD drawn on a rectangular page PQRS is
represented alongside.
A
P
Q
The length of line segment AC is given by x and the
length of line segment BD is given by y.
5.1
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
5.2
If you were to cut out the shape of the kite from
the page, what percentage of the paper would you
discard?
Show all working.
(3)
5.3
If D and B were the midpoints of PS and QR
respectively, what would be the shape formed?
(2)
S
C
5.4
Your Maths Teacher made a statement in class, “A kite that has equal diagonals
is a rhombus.” Is their statement true or false? Justify your answer by either
using a proof if the statement is true or a counter example if the statement is
false.
(3)
[10]
1.1
Complete the following sentences by writing down the missing word:
1.1.1 A quadrilateral whose diagonals are equal but which is not a
square must be a …………… (1)
1.1.2 If a parallelogram has perpendicular diagonals then it must be a
R
………..
(1)
C
D
For quadrilateral ABCD it is given that B̂  D̂
1.2
and that AC bisects BĈD
What kind of quadrilateral is ABCD? (1)
[3]
A
B
y
x
x


In the above isosceles triangle, each base angle is x and the apex angle is y .

If the apex angle is 15 larger than the base angles, find the size of each angle.
[6]
P
7. 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)
5.1
State whether the following conjectures are true or false. If false, give a reason for
your answer.
5.1.1
5.1.2
5.1.3
5.1.4
All regular pentagons have 5 sides.
All octagons are similar.
Triangles are polygons.
All polygons are convex.
(6)
9.1
In the sketch below, BA // QT and AP = AC. BCˆ Q  45 and ACˆ T  55
A
x
T
55
B
P
45
C
Q
Calculate the size of x , showing all working and giving reasons. (8)
9.2
Given that AD = DB and AE = EC, what can be said about DE and BC in the sketch below?
(2)
A
D
B
>>
>>
E
C
9.3
In the sketch below AD = AS = SR, AE // SM and RE // PM.
Prove:
10.3.1 DR = RP
(4)
10.3.2 AD 
1
DP (2)
6
D
A
S
R
P
[16]
E
>>
>>
M
7.1
Calculate the value of x and y, giving reasons:
W
x
S
10
24
T
25
y
2,8
R
V
(5)
7.2
Find the values of a, b and c, giving reasons:
P
b
Y
38°
X
9cm
74°
a
R
c
(7)
Q
7.3
reasons:
If AE
Prove that ΔABC is isosceles, giving
BC and E is the mid-point of BC.
A
B
1.1
1.1.1
E
Find the values of the variables giving reasons:
C
(6)
50 
70 
b
c
(4)
1.1.2
A
B
50 
d
e
C
O
D
OB // DC
Ô is the centre of the circle.
(4)
1.1.3
G
13
f
H
12
J
(3)
1.1.4
h
g
(4)
[15]
1.2
Are these lines parallel or not? Give reasons.
1.2.1
AB and CD
A
B
30 
D
C
150 
(2)
1.2.2
DE and NR
M
65
D
E
140
N
R
(2)
1.2.3
QR and ST
P
3
Q
4
R
5
S
8
T
(2)
[6]
1.3
The following pairs of triangles are similar, find the unknown sides. Show all working.
MNO  PQR
M
P
10
5
a
8
N
b
O
Q
7
R
[4]
1.4
A
P
O
R
B
M
C
If AR = 8 units, BP // MR, AO = OM and BM = MC ,find the length of RC.
Give reasons for your statements.
[4]
1.5
M
N
P
Q
1.5.1 There are many ways to prove a quadrilateral is a parallelogram.
Four of these are
1. both pars of opp. sides are equal.
2. one pair of opp. sides are equal and parallel.
3. both pairs of opp. Sides are parallel.
4. diagonals bisect.
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)
1.5.2
Explain with reasons why?
1
NO =
NB.
3
(2)
QUESTION 1
1.1
Refer to Figure 1.
A
y
y
AB ∥ PQ
M
AM bisects BÂP and PM bisects AP̂Q
x
1.1.1
Write down, with a reason, the numerical value
B
P
x
Q
Figure 1
of
2x  2 y
1.1.2
(1)
Calculate the size of AM̂P (2)
QUESTION 2
Refer to Figure 3.
The circle with centre O has a chord AB.
O
OP is drawn perpendicular to AB.
Figure 3
A
2.1
2.2
Prove that ∆OAP  ∆OBP (4)
If AB = 8 cm and OP = 3 cm, determine the radius of the circle.
[7]
B
P
(3)
QUESTION 1: Geometry
1.1
Write down (without reasons) the size of the angles marked x and y in the following:
(a)
(b)
y
x y
32°
x
(4)
1.2
(a)
Work out the value of x in the following. Reasons must be given.
(b)
(c)
x
x-24°
53°
x
3x
62°
(6)
QUESTION 1. 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. In the figure, PQRS is a parallelogram with
diagonals PR and SQ intersecting at O.
Find the value of x (giving reasons).
P
Q
6x
4x
80
O
S
R
[5]
QUESTION 3. 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 .
(3)
(7)
(4)
2,5 cm
x
A
1.
b
a)
B
name a six sided polygon.
[14]
(1)
b)
c)
1.
Find the size of an interior angle of a regular octagon.
If the two shorter sides of a right angled triangle are 5 cm and 12 cm,
find the length of the hypotenuse.
(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.
b)
If a quadrilateral has two right angles and two equal diagonals, then it is a rectangle.
2.
In quadrilateral ABCD , E is the midpoint of AB , F is the midpoint of CD and
BC  AD
. 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.
E
A
B
H
D
Investigate (Cross’s Theorem):
F
G
C
If two rectangles have the same diagonal, must the rectangles be congruent?
Answer: No
QUESTION 5:
After the studying the following diagram, complete the statements below. (no reasons necessary).
F
12
B
H
3
2
11
1
G
9
I
6
4
7
C
5
5.1
2 + 6 + 4 = . . . . . degrees
5.2
5=2+ .....
5.3
1 + 7 + 9 + 11 = . . . . . degrees
5.4
3 + 5 + 8 + 10 + 12 = . . . . . degrees
5.5
1 + 3 + 5 + 7 + 8 = . . . . . degrees
8
J
10
E
A
(10)
QUESTION 6:
6.1
Prove the theorem that states that the internal angles of a triangle
are supplementary.
6.2
D
Find the value of x in the diagram below if AB=AC and BC=BE.
A
34
F
(5)
(6)
QUESTION 7:
7.1
List three properties of a rhombus that are not properties of a
parallelogram.
(3)
7.2
PQRS is a rhombus with PS = PR = x mm. Express the following
lengths in terms of x . No reasons are required.
7.2
7.2.2
7.2.3
.1
QT
QS
PT
P
Q
T
S
7.3
R
(5)
In the figure, KMQR is a rhombus. The bisector of MKˆ Q cuts
MQ at P. Prove that MPˆ K  3MKˆ P
K
M
P
(6)
QUESTION 8:
Q
R
8.1
List 5 ways in which you could prove that a quadrilateral is a
parallelogram.
(5)
8.2
Prove any one of these, except the one that is the definition of a
parallelogram.
(5)
8.3
ABCD is a quadrilateral with AD//BC and E on BC such that
AB = AE and EC =CD.
A
D
2x
x
8.3.1
Prove that ABCD is a parallelogram.
(6)
8.3.2
Prove that EA=EC.
(2)
8.3.3
Calculate x if AEˆ D  105

(4)
QUESTION 2:
2.1
State the four cases of congruency.
(4)
2.2
In the diagram ABCD is a parallelogram and AD  DE .
Prove that DF  FC
B
A
D
C
F
(7)
QUESTION 3:
E
3.1
Redraw these sketches of a rhombus on your answer sheet and,
using appropriate symbols, show all the properties of a rhombus.
(6)
3.2
List five ways in which one can show that a quadrilateral is
a parallelogram.
3.3
(5)
F
In the diagram alongside, FGHJ is
a rhombus. FO  JK and OG  KH .
G
O
J
H
K
3.3.1
Prove that JOHK is a parallelogram.
(5)
3.3.2
Now show that JOHK is a rectangle.
(2)
QUESTION 4:
4.1
Using the diagram below, prove the theorem that the diagonals of
a parallelogram bisect each other.
M
N
O
Q
4.2
P
(6)
In the diagram RSTV is a parallelogram
Y
R
S
O
V
W
T
4.2.1
Prove that ROY  TOW .
(5)
4.2.2
Prove that RYTW is a parallelogram.
(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.
Question 1
1.3
Find the values of the variables giving reasons:
1.1.1
50 
70 
b
c
(4)
1.1.2
A
B
50 
d
e
C
O
D
OB // DC
Ô is the centre of the circle.
(4)
1.1.3
G
13
f
H
12
J
(3)
1.1.4
h
g
(4)
[15]
1.4
Are these lines parallel or not? Give reasons.
1.2.1
AB and CD
A
B
30 
D
C
150 
(2)
1.2.2
DE and NR
M
D
65
140
N
E
R
(2)
1.2.4
QR and ST
P
3
Q
4
R
5
S
8
T
(2)
[6]
1.3
The following pairs of triangles are similar, find the unknown sides. Show all working.
MNO  PQR
M
P
10
5
a
8
N
b
O
Q
7
R
[4]
1.4
A
P
O
R
B
M
C
If AR = 8 units, BP // MR, AO = OM and BM = MC ,find the length of RC.
Give reasons for your statements.
[4]
1.5
M
N
P
Q
1.5.3 There are many ways to prove a quadrilateral is a parallelogram.
Four of these are
1. both pars of opp. sides are equal.
2. one pair of opp. sides are equal and parallel.
3. both pairs of opp. Sides are parallel.
4. diagonals bisect.
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)
1.5.4
Explain with reasons why?
1
NO =
NB.
3
[6]
Question 2
(2)
y
B (3; 7)
1
x
A (2;3)
Show all working and use above sketch to:
2.1.1
Find the distance between A and B.
(3)
2.1.2
Find the gradient from A to B.
(2)
2.1.3
Find the midpoint between A and B.
(3)
2.1.4
Write down the equation of the straight line AB.
(2)
2.1.5
Which of the following points will be on the line AB. Show all working.
a) (4;9)
b) (2; 5)
(2)
(2)
[14]
2.2
Monique works in the post office and can stamp 750 letters every 5 minutes.
2.2.1
Calculate the average rate of change at which she stamps the letters.
(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)
2.2.3 Find an equation that links the number of stamps (y) to the minutes (x) and
draw this graph on a suitable axis.
(4)
[8]
2.3
Draw sketch graphs if:
2.3.1 m is negative and c is positive.
(2)
2.3.2 m is positive and c is negative.
(2)
(No values are to be put onto these graphs, indicate only the x and y axis.)
[4]
2.4
P (2; 5)
Q (5; 7)
T
S
R (4;1)
PQRS is a parallelogram. Remember that in a parallelogram, diagonals bisect.
2.4.1 Find T, the intersection of the two diagonals.
(2)
2.4.2 Find S, one of the vertices of the parallelogram.
(2)
[4]
2.5
Mr Boon is currently driving a Toyota Prado which cost him R690 000
new, in January 2004. By the end of each year the car’s value depreciates
such that it forms a linear equation. By the end of 2006 the value of the
car is R640 000.
A new Toyota Verso increases in price each year also such that it forms
a linear equation. The price of a Verso was R240 000 in the beginning
of 2006 and will cost R270 000 at the beginning of 2007.
2.5.1 In each case find suitable equations and plot them on one Cartesian plane. (2)
2.5.2 Mr. Boon wants to drive his Prado has long as possible but want to buy a new verso without
having to pay in. Find the point at which Mr Boon can sell his Prado and buy a brand new Verso without
having any money left.
(3)
[5]
Total: [70]
Sw/rl/24.04.06
1. 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.
T
I
Calculate the size of x.
1 2
1
K
x
2
N
F
(4)
b)
X
T
A
B
EXTEND is a regular.
hexagon. AN // BP.
Calculate the value of a.
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.
B 72
72 E
74
74
C
D
(3)
b) State whether the following are true or false. If false, explain why.
i) If Pˆ  Qˆ , then the supplement of P̂ is less than the supplement of Q̂ . (2)
ii) In ABC, Aˆ  90, AB = 3, BC = 4 and AC = 5.
(2)
c) If ABC  XYZ, then Aˆ  Xˆ , Bˆ  Yˆ , Cˆ  Zˆ .
i) Write down the converse of this statement.
ii) Is this converse true or false? If false, give a counter example.
(1)
(2)
4.
A
1
2
G
1
2
H
F
1
56
2
66
1
2
3
1
B
C
3
D
E
Aˆ1  Dˆ 1  x ; Cˆ 2  y ; Dˆ 3  z. AC  CD and CH  HD.
Calculate the values of x, y and z.
5a)
L
Ŝ  L̂ ; Ĝ  Â.
T is the midpoint of SL.
G
N
Is SG = TA?
Show all reasoning.
T
R
Y
(7)
b)
A
B
C
2,42
AB = AC and  = 90. Calculate AB.
(3)
6.
9,45 cm
10,5
cm
Above is a picture of the Arc de Triomphe in Paris. Alongside is a plan
of the front, with the measurements of the height and width. If the Arc
de Triomphe is 50m high, how wide is it? Show all working.
(3)
7a)
A
AB and AD are each divided into
four equal parts.
P
S
Ĉ  D̂.
Determine the length of:
i) BD
Q
T
ii) PS
R
U
B
D
(6)
48
C
b)
T is the centre of the circle.
PQ // ST. If PS = 3cm,
determine the length of SR.
R
S
P
T
Q
(4)
QUESTION 4
ABCD is a rectangle. O is any point inside
ABCD.
B
A
Prove that :
OAB + OCD = OAD + OBC
O
D
C
QUESTION 1
Complete the following statements:
1.1
A trapezium is …
(2)
1.2
A rhombus is …
(2)
1.3
A kite is …
(2)
1.4
RSTU is a quadrilateral with  R = 4x,  S = 5x,  T = x and  U = 2x.
1.4.1
Calculate the size of  S.
1.4.2
Prove that RSTU is a trapezium.
(3)
(5)
[14]
QUESTION 2
A
ABCD is a parallelogram with  A = 68 .
EC bisects  C. Calculate the values
of p and q.
E
o
B
q
o
68
p
[5]
C
D
QUESTION 3
3.1
Write down the definition of a parallelogram.
3.2
PS // QR. T is the midpoint of PR.
Prove that PQRS is a parallelogram.
STQ is a straight line.
(2)
P
Q
T
S
(6)
[8]
R
QUESTION 4
U
V
4.1
Prove the theorem that states that the
opposite angles of a parallelogram
are equal. Redraw the diagram.
X
4.2
DEFG is a parallelogram
with GH // IE.
H
Prove that DI = FH.
D
W
(6)
E
I
G
F
(7)
[13]
QUESTION 5
Parallelogram ABCD is shown with
AX bisecting  A and DX bisecting
 D.
Prove that  AXD is a right angle.
A
B
X
D
C
[5]
1.
Find a , b, c and d , giving reasons :
A
M
L
B
1.1
1.2
a
E
C
Q
130
20
D
O
N
P
(2)
(5)
/7/
QUESTION 2
2.
F
Find value of x , giving reasons.
A
2.1
D
190x
x
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 :
3.1.
2 angles each equal to x
(2)
D
3.2
2 angles each equal to 2x
3.3
(2)
Hence calculate the value of x .
(2)
B
/6/
A
QUESTION 4
4
Find angle AOC , giving reasons.
/4/
C
D
QUESTION 5
5
5.1
Prove ABC  AED
(3)
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.
Add all information given in the question to the diagram.
Support all statements with reasons - using the accepted abbreviations
Present your argument succinctly (in a brief, concise, logical manner)
Parallel Lines.
[a = b (vert opp ‘s)]
a = d (corr ‘s, PQ // RS)
b = d (alt ‘s, PQ // RS)
b + e = 180o (co-int ‘s, PQ // RS)
P
a
c
Q
b
R
d
e
S
Calculate, with reasons, the value of x in each of the following cases:
1
A
E
B 3x
C
2
A
D
F x + 50o
G
AB // EF & BD // FG
3
B
x + 14o
2x + 10o
C
D
A
AB // CD
B
4
A
B
o
o
125
C
50
x
C
D 160o
E
D
AB // DE
Solve questions 3 and 4 in two different ways!
x
E
AB // DE
Triangles
The angle sum of any triangle is 180o
The exterior angle of any triangle is equal to the sum of the two interior opposite angles.
An isosceles triangle has two equal sides, and the angles opposite these sides are also equal.
Find, with reasons, the value of x and y in each of the following:
5
A
64o
6
A
F
x y
3x y
B C
AB // FE
AC = AD
o
x
84
D
38o
E
B
D
C
A
E
7
8
34o
Prove, with reasons, that AC = CD
if AB = BC
0
136
E
B
AB // DC.  DAB = 120o ,
DBC = 85o,  DCB = 65o
Prove that  ADB is isosceles.
A
D
C
B
D
C
9
PQRS is a parallelogram.
Prove that  P =  R.
P
Q
S
10
In  PQR,  P =  Q = x
RT // QP, QR is produced to S.
Prove that RT bisects  PRS.
R
P
T
Q
11
R
BD = DC = DA
 ABC = x,  BAC = y
Find the size of  ACB in degrees.
S
A
D
B
y
x
C
A
12
AB = BC. AD = AC.  BAD = 30o
and  ABD = x.
Calculate the size of x in degrees.
30o
x
B
13
D
 SPR =  SRP = x
 QRP = 90o
Prove that S is the midpoint
of PQ.
P
S
x
x
Q
14*
C
EC = BC.  ABC =  BDC = 90o.
Prove that BE bisects  ABD.
R
A
E
D
B
C
Congruency
Two triangles are congruent if they are identical in shape and size.
(Two triangles are similar (///) if they are have the same shape - angles correspondingly equal and
sides in proportion - but not necessarily equal in size. Tested using the “ test”)
Two triangles can be proven to be congruent by proving one of the following sets of conditions.
To ensure equal size, at least one length must be the same for each triangle.
1
S.S.S
2
S. included . S
3
S. . .
4
90o. H. S.
Approach:
Always state the triangles in which you are working - labelled in corresponding order.
State the 3 conditions with reasons, keeping information under the correct triangle.
State that the triangles are congruent (  ) and add the condition for congruency used.
Then use the congruency to draw any further conclusions required.
eg
Prove that AB = CD given AB // CD
and BO = OC.
A
B
O
In  ABO and  DCO
1
BO
= CO
2 A = D
3 O = O
 ABO   DCO
 AB = CD 
(given)
(alt ‘s, AB // CD)
(vert opp ‘s)
(S. . .)
C
D
B
A
15
O is the centre of the circle.
 AOB =  COD
Prove that AB = CD.
O
C
D
16
AD = BC and  EDA =  FCB
Prove that  ACD =  BDC.
A
B
O
E
17*
x
D
x
C
AB = AC and EB = EC
Prove that AD  BC
F
A
E
P
B
C
D
18*
PS = PT and PQ = PR.
Prove that  QPO =  RPO.
T
S
O
Q
R
2.2
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. Give your reasons clearly.
A
B
C
D
5.1
145 o
x
115 o
y
E
(4)
5.2
A
65 o
25 o
D
5.3
y
B
C
M
54 o
A
x
N
x
2x
C
B
y
2y
O
5.4
(4)
P
D
(4)
A
40 o
70 o E
x
C
6
D
Write down the 4 reasons for congruency.
(4)
(2)
QUESTION 1
1.1
Write down the definition of a parallelogram.
(2)
1.2
For each of 1.2.1 to 1.2.3 below, write down one property of a parallelogram
which has to do with its ...
1.2.1 sides (1)
1.2.2 angles (1)
1.2.3 diagonals
(1)
1.3
Sketch a rhombus on your answer sheet. Make markings on your sketch which
indicate at least 5 properties of a rhombus. (Think of sides, angles, diagonals,
lines of symmetry, etc.)
(5)
/10/
QUESTION 2
Use the properties of these quadrilaterals to find your answers. Reasons need not be given in this
question.
2.1
2.2
2.1.1 Name the quadrilateral.
(1)
2.1.2 Write down the sizes of angles a and b .
(2)
2.2.1 Name the quadrilateral.
(1)
2.2.2 Write down the sizes of angles x and y .
(2)
/6/
a
b
x
y
95
QUESTION 3
From this point on, you should give clear reasons for all your steps.
A
3.1
In ABC, A = 75 and C = 60.
DEFB is a parallelogram.
Calculate the angles of parallelogram DEFB.
D
(6)
E
B
C
F
3.2
Calculate the angles of
parallelogram PQRS.
P
6x  12
(4)
2x + 40
3.3
The drawing shows parallelogram
R ABCD and parallelogram ABEC. Prove
S that
C is the midpoint of DE. (‘Midpoint of DE’ means it lies in the middle 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. Show full proofs, giving reasons wherever necessary. There are 8 questions.
K
J
Find the size of  a.
1
F
a
H
G
L 30o M
[4]
T
2
TR = TS. QRS is a straight line.
Find the size of  b.
b
=
=
110o
Q
R
S
[4]
D
3
o
EF // LK // GHJ.
Find the sizes of  c,  d and  e.
50
E
F
L
K
3d – 20o
e
c
d
G
J
[6]
H
In  ABC, P is any point on AB.
BT bisects  ABC. PT // BC.
Prove, with reasons, that PT = PB.
4
A
P
o
T
o
B
C
[4]
P. T. O.
X
 XYZ is drawn together with four other triangles.
Just name, in correct order, the one triangle that is
congruent to  XYZ and give the reason for the
congruency. Lengths are in cm.
5
3
38o
Y
A
D
92o
38o 50o
C
B
F
4
G
K
3
38o 50o
3
E
92o 2,5
50o Z
J
4
50o
4
2,5
[2]
H
M
3
L
A
\
6
AB = DC and  ABC =  DCB.
Prove, using congruency, that
 ABC   DBC and that
 A =  D.
B
C
/
D
[5]
P
7
PQT and PSR are straight lines.
PS = PQ.
=
Prove that TS = RQ.
Q
T
x
=
x
S
R
[5]
1.
P
In the diagram alongside
 Q = 63°,  P = 85°,
and TSRU is a parallelogram.
Calculate the sizes of the
angles of the parallelogram,
giving reasons for all statements.
T
S
Q
R
U
(8)
2.
K
In parallelogram KLMN
 K = x+60° and
 M = 3x-30°.
L
Find the sizes of the angles
of the parallelogram.
N
M
(6)
3.
In parallelogram DEFG
 D = 4x and  E = 2x-30°.
Find the sizes of the angles
of DEFG.
E
D
G
F
(6)
Grade 10 SG
1
GEOMETRY: CONGRUENCY RIDERS
Prove that AO = OD and BO = OC
if AB = CD and AB // CD.
A
1999
B
O
C
D
A
2
B
Prove that AB = CD and AB // CD.
O
C
3
O is the centre of the circle.
Prove that AC = CB.
D
O
A
C
B
A
4
D
Prove that AC = BD
=
A
5
AB = AE.
Prove that CE = BD.
B
=
x
O
B
=
x
=
C
E
O
C
6
D
AB = AE, AD = AC and  BAE =  DAC
Prove that BC = DE.
A
/
x
=
=
B
7
Prove that AB = DC
and AB // DC.
A
=
=
D
x
/
E
B
C
D
C
In each question, redraw the diagram in your workbook, and give reasons for all the statements that you
make.
P
1.
PQRS is a parm with  P = 60°, PQ = PS
and QS = 50 mm.
1.1
1.2
Q
Prove that PQRS is a rhombus.
Find the perimeter of the rhombus.
S
R
A
2.
In  ABC,  A = 80° and  C = 55°.
Calculate the angles of the parallelogram
DEFB.
D
E
B
3.
K
Calculate the angles of the parm KLMN,
if  N = 5x – 12° and  L = 3x + 18°
N
L
M
A
4.
ABCD is a parm, with EF AB,
GH BC, and  B = 54.
Calculate the sizes of
i)  D
ii)  A
iii)  AGK iv)  EKH
E
G
D
K
B
5.
ABCD is a trapezium with AD BC,
and  A =  D = 50°. If FC = DC,
prove that ABCF is a parallelogram.
C
F
H
C
F
F
A
B
D
C
1.
ABDE is a parm. AF = CD.
A
F
Prove: 1.  ABF   DEC
G
2. FG = GC
2.
E
B
D
C
A
AO = OC;  BAC =  ACD;
 ABD =  BDC
D
O
Prove that ABCD is a parm.
B
C
F
3.
BA = AF = CD; FE = EC;
AE = ED
Prove that ABCD is a parm.
E
A
D
C
B
A
4.
F
ACDF is a parm.
Prove 1. BF = CE
2. BCEF is a parm
3.  BCE =  ABF
B
E
C
D
A
5.
F
D
AGEF and ABCD are parms.
Prove that  GEF =  BCD
G
E
B
C
1.
Complete the following sentences, writing the words needed to complete the sentence
P
grammatically in the space provided on the answer sheet:
1.1
S
The definition of a parallelogram is as a quadrilateral with
55 mm
opposite sides …..….
V
A rhombus is a ……. with ……….. 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 ………..
R
If a quadrilateral has two pairs of adjacentQ
sides equal and opposite sides not equal it is a
1.2
1.3
1.4
1.5
1.6
………
The diagonals of a rhombus ………… each other and are ……………
(10)
1.7
2.
From the sketch alongside, name
(giving the vertices in alphabetical order)
2.1
2.2
2.3
3.
one rhombus
three parallelograms
two trapezia (6)
For each of the sketches find the size of the angles marked with a small
letter. Fill in your answers (no reasons required) on the answer sheet.
C
3.1
3.2
D
B
A
F
E
(7)
h
A
4.
a PQRS is a rhombus. 58°
In the figure,
D
22°
X
RT bisects QR̂P .
Give, with reasons and on the
answer sheet,
B
b
4.1
the size of SV̂R
4.2
the size of VR̂T
4.3
the length of SQ
c
80°
d
C
(6)
U
k
m
V
122°
31°
E
W
5.
Calculate the areas of each of the quadrilaterals shown. Show your working.
5.1
5.3
B
A
C
900 m
2
20 m
F
E
D
5.2
5.4
A
B
17 mm
30 mm
(11)
40 mm
C
D
N
M
P
35 mm
QN = 20 cm
S
Q
QK = 30 cm
10 mm
R
Q
P
K
F.
For each of the following figures, make an equation involving x, giving reasons, and then solve it
to find the value of x.

B
A
x + 21
O
D
17
3x – 18
C
Parallelogram 
48
65
Parallelogram 
63
Rhombus 
53
Rectangle 
37
27
Kite 
1.
State whether the following sentences are TRUE or FALSE. For those which are
FALSE, give a corrected version.
1.1
1.2
1.3
1.4
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)
2.1
Give four properties of a quadrilateral such that each one on its own would guarantee
that the quadrilateral is a parallelogram.
(4)
2.2
What extra properties do you need to prove that a parallelogram is a square?
(2)
2.3
Name two quadrilaterals whose diagonals are perpendicular
(2)
A
3.
ABCD is a square, with BF̂D = 125.
Find, giving reasons clearly, the size
of BT̂C
B
T
F 125
4.
In the figure, AED and BFC are
straight lines.
(5)
C
D
E
A
D
115
65
Prove that ABCD is a parallelogram.
65
B
C
F
(6)
X
5.
The diagonals of parallelogram ABCD meet in O.
DB is produced to X and YC is drawn parallel to
AX to meet BD produced at Y.
A
B
O
D
C
Y
5.1
Prove that  AXO   CYO
(4)
5.2
If CX and AY are joined, prove that AXCY is a parallelogram.
(3)
Question 1.
State whether the following statements are true or false:
1.1
In a trapezium both pairs of opposite sides are parallel.
1.2
The diagonals of a kite bisect each other at 900.
1.3
If a parallelogram has equal diagonals then it must be a square.
1.4
A square is always a rhombus, but a rhombus is only sometimes a square.
1.5
If the diagonals of a quadrilateral are equal then it must be a rectangle.
[5]
14]
Question 3.
STRV and PURV are parallelograms.
U
P
S
T
V
R
Prove that PSTU is a parallelogram.
[6]
Question 4.
4.1
Write down the ratio of the areas of the figures specified.
Reasons and steps are not required.
ΔBDE : ΔABC
ΔAEC : parmABCD
A
4·1·1
4·1·2
E
D
A
D
E
B
C
B
C
(4)
4.2
The area of parallelogram
ABCD=1500mm2 and
BC=30mm.
Calculate:
B
A
G
E
D
F
C
4.2.1 the length of EF
4.2.2 the area of ΔGDC
4.2
(6)
ABCD is a parallelogram.
Prove that Area ΔBCP  Area ΔABQ
(5)
D
A
B
P
C
Q
1.
State whether the following sentences are TRUE or FALSE. For those which are
FALSE, give a corrected version.
1.1
1.2
1.3
1.4
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)
2.1
Give four properties of a quadrilateral such that each one on its own would guarantee
that the quadrilateral is a parallelogram.
(4)
2.2
What extra properties do you need to prove that a parallelogram is a square?
(2)
2.3
Name two quadrilaterals whose diagonals are perpendicular
(2)
B
A
1.
51
In the figure ABCD is a parallelogram.
Prove that it must be a rhombus
M
39
D
C
B
A
2.
In the figure ABCD is a parallelogram.
Prove that it must be a rhombus.
43
94
D
3.
C
B
A
In the figure ABCD is a parallelogram.
ˆC
Prove that ABˆ C  BM
D
C
6.
ABCD is a rhombus, and DC is extended to P so
that BP = BC.
If DBˆ C = x,
find the following in terms of x:
4.1
DAˆ C
4.2
P̂
D
M
B
A
x
M
C
P
7.
ABCD is a parallelogram.
B
A
Prove
P
5.1
 ADM   CBP
5.2
DP = BM
5.3
MAˆ B  PCˆ D
M
D
C
Q
A
8.
B
ABCD is a parallelogram.
Prove
9.
6.1
 ADP   CBQ
6.2
AQ = PC
D
C
P
Is AQCP a parallelogram? Give reasons for your answer
A
ABCD is a parallelogram.
DP = BQ.
B
Q
Prove that AQCP is a parallelogram.
P
D
C
B
A
P
10.
ABCD is a parallelogram, and DP  AC
with BQ  AC.
Q
Prove that BPDQ is a parallelogram.
D
C
B
A
11.
ABCD is a parallelogram, and
DP  QB. Prove that DPBQ is
a parallelogram.
P
Q
B
A
12.
ABCD is a parallelogram, and
DP = PV while BD = QV.
Q
V
Prove that APCQ is a parallelogram.
P
D
13.
ABCD is a parallelogram.
(HARD)
C
U
B
A
Given that DP = PU and AP = PV,
prove that BV and CU bisect
each other.
Q
P
V
D
C
QUESTION 4
[3]
Refer to the diagram below, and solve for x (you must show all reasons).
QUESTION 5
[8]
Refer to the diagram below, and answer the following questions (with reasons).
5.1)
Determine the size of P (in terms of x).
(3)
5.2)
If T 2  90  x , determine the size of R (in terms of x).
(2)
5.3)
If PTR=110 , determine the value of x.
(2)
5.4)
What type of triangle is QTR ?
(1)
QUESTION 6
Prove ACD  CAB
[4]
QUESTION 7
[4]
Based on the information provided, make a conjecture about the quadrilaterals below. Then prove your
conjecture.
You must give the best classification possible:

If a quadrilateral is a rhombus, it is not good enough to classify it as a parallelogram.

If a quadrilateral is a rectangle, it is not good enough to classify it as a parallelogram.

If it is a square it should not be classified as a rectangle.
7.1)
(2)
7.2)
QUESTION 8
(2)
[11]
In the diagram below AD=AG and EF||DG
8.1)
If ADG = x , State (with reasons) three other angles equal to x.
(3)
8.2)
Prove that ED = FG.
(2)
8.3)
Prove that EDG  FGD .
(3)
8.4)
Name 2 other pairs of congruent triangle (do not prove they are congruent)
(2)
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.
In the diagram, AD is an angle bisector of CAB
perpendicular to AD.
Prove that
AE AC

FB CD
question until you have