New Century Math Yr 9 - Chapter04 Investigation Geometry
New Century Math Yr 9 - Chapter04 Investigation Geometry
New Century Math Yr 9 - Chapter04 Investigation Geometry
Investigating
geometry
The word geometry comes from the Greek work geometria, which means land
measuring. The principles and ideas of geometry are evident in many aspects of our
lives. For instance, geometry may be used in the design of buildings, bridges, and
patterns for tiles and wallpaper. It may even be used when playing billiards and
pool. Many examples of geometrical patterns can be seen in nature.
The contact lens in this photo combines geometry, algebra and biology to correct
problems with vision.
04_NC_Maths_9_Stages_5.2/5.3 Page 103 Friday, February 6, 2004 2:12 PM
Wordbank
■ geometry The branch of mathematics that deals with the measurement,
properties and relationships of points, lines, surfaces and solids.
■ closed figure A plane shape that completely surrounds an area. Squares
and circles are closed figures but an angle is not.
■ polygon A plane closed figure or shape with straight sides. A regular
polygon, such as a square, has all its angles equal and all its sides equal.
■ diagonal An interval joining a vertex to another non-adjacent vertex.
■ exterior angle The angle formed outside a polygon when one of its sides
is extended.
Think!
A square can be considered to be a rhombus but a rhombus cannot be
considered to be a square. Explain why. Can a square also be considered to
be a rectangle?
Start up
1 Find the size of each angle marked by a pronumeral, giving reasons.
Worksheet
a b c
4-01
Brainstarters 4 27°
110°
Skillsheet
4-01 c°
x° 37°
Types of angles m°
Skillsheet
4-02 d e f
30° 35°
Measuring
angles h° 42°
h°
139° y° 40°
70° c° x° x°
v°
p° 55° a°
35°
d e f
Worksheet
4-02
h° 140°
A page of angles b° w° 30°
h° c°
85° y°
65° q°
3 Find the value of each pronumeral, giving reasons for each step you use.
a b c
Skillsheet
4-03 80° 120°
3k°
Angles and
parallel lines 66°
5m°
4w°
d e f
Worksheet
(5m + 30)°
4-03 75° 4k°
Find the missing
angle 1 3c° 70°
60°
Skillbank 4
Finding percentages of multiples of 100
Since a percentage is a number out of 100: SkillTest
4-01
• 1% of 100 = 1 • 1 1--- % of 100 = 1 1--- • 2% of 100 = 2, and so on. Finding a
2 2
percentage of a
1 Examine these examples: multiple of 100
a Find 1% of 300. b Find 4% of 500.
1% of 100 = 1 4% of 100 = 4
∴ 1% of 300 = 1 × 3 (since 300 = 100 × 3) ∴ 4% of 500 = 4 × 5 (since 500 = 100 × 5)
=3 = 20
2 Find the following amounts.
a 1% of 600 b 3% of 400 c 5% of 200 d 2% of 300
e 4% of 600 f 3% of 800 g 6% of 400 h 2% of 900
i 8% of 400 j 4% of 700 k 3% of 1200 l 13% of 200
3 Examine these examples:
a Find 1 1--- % of 200.
2
1 1--- % of 100 = 1 1--- or 1 1--- % of 100 = 1.5
2 2 2
∴ 1 1--- % of 200 = 1 1--- ×2=3 1 1--- % of 200 = 1.5 × 2 = 3.0 or 3
2 2 2
b Find 3 1--- % of 500.
2
3 1--- % of 100 = 3 1--- or 3 1--- % of 100 = 3.5
2 2 2
∴ 3 1--- % of 500 = 3 1--- ×5 ∴ 3 1--- % of 500 = 3.5 × 5
2 2 2
= 3 × 5 + 1--- × 5 = 17.5
2
= 15 + 2 1--- = 17 1---
2 2
4 Find the following amounts.
a 1 1--- % of 400 b 4 1--- % of 200 c 2 1--- % of 300
2 2 2
d 3 1--- % of 300 e 1 1--- % of 400 f 4 1--- % of 500
2 2 2
g 2 1--- % of 700 h 3 1--- % of 400 i 1 1--- % of 800
2 2 2
5 Examine these examples:
a Find 1.5% of 400. b Find 2.25% of 300.
1.5% of 100 = 1.5 2.25% of 100 = 2.25
∴ 1.5% of 400 = 1.5 × 4 = 6.0 (or 6) ∴ 2.25% of 300 = 2.25 × 3 = 6.75
6 Find:
a 1.25% of 300 b 4.25% of 200 c 2.75% of 500
d 1.3% of 400 e 4.2% of 300 f 2.5% of 1200
Properties of triangles
A triangle is a three-sided figure. Triangular shapes are
used in many everyday objects such as:
• the frames of bicycles and motorcycles
• the roof trusses of houses
• the sails of boats, sailboards and windsurfers
• the exteriors of many buildings
• some road signs.
SkillBuilder
23-02
Introduction to
shapes – Part II
scalene isosceles equilateral
Summary
The properties of the various types of triangles can be summarised as follows:
Scalene triangle – no two sides equal
– no two angles equal
– no symmetry
Isosceles triangle – two equal sides
– the angles opposite the equal sides are equal
– one axis of symmetry
Equilateral triangle – all sides equal
– all angles equal to 60°
– three axes of symmetry
– has rotational symmetry
b° a° + b° + c° = 180°
c°
Proof: U
P
Consider any triangle PQR in which the angles are a°, b°
and c°. Construct a line through P parallel to QR. a° V
∴ ∠UPQ = b° (alternate angles, UV QR)
b°
and ∠VPR = c° (alternate angles, UV QR) Q
c°
R
but ∠UPQ + ∠QPR + ∠VPR = 180° (angle sum of a straight line)
∴ a° + b° + c° = 180°
So the sum of the angles of a triangle is 180°.
Working mathematically
Reasoning: The exterior angle of a triangle
1 a Draw ∆UTV and produce V The side UT has been
(extend) the side UT to P, produced to P.
as shown in the
diagram. ∠VTP is an exterior
b Use a protractor to angle of ∆UTV.
measure ∠VUT
U T P
and ∠UVT.
c Measure ∠VTP. What do you notice?
2 a Repeat the procedure in Question 1 for ∆ABC which has the side AB produced to
point D.
b Measure the exterior angle CBD and then measure the angles CAB and BCA. What do
you notice? (The angles CAB and BCA are referred to as the interior opposite
angles to angle CBD.)
3 Write a conclusion for your results in Questions 1 and 2.
m°
Y p°
p° = m° + n° Z
W
Proof: X
Consider any triangle XYZ in which the angles are m°, n°
n°
and k°. Produce the line YZ to the point W.
m° + n° + k° = 180° (angle sum of a triangle)
∴ m° + n° = 180° − k°
but ∠XZW = 180° − k° (angles on a straight line) m°
Y
∴ ∠XZW = m° + n° k° p°
∴ p° = m° + n° Z
W
or A
Consider any triangle XYZ in which ∠YXZ = n° and X
∠XYZ = m°. Construct a line AB through X parallel n° B
to YW.
∴ ∠AXY = ∠XYZ = m° (alternate angles, AB YW)
∴ ∠AXZ = m° + n°
m°
but ∠AXZ = ∠XZW (alternate angles, AB YW) Y p°
∴ ∠XZW = m° + n°
Skillsheet Z
∴ p° = m° + n° W
4-04
Starting
Geometer’s Using technology
Sketchpad
Angle sum of a triangle
Skillsheet Step 1: Construct a triangle using your drawing program.
4-05 Step 2: Measure its three angles.
Starting Cabri Step 3: Add the three angles. What is the sum?
Geometry Step 4: Construct two more triangles, measure their angles and determine the sum.
What can you say about the angle sum of a triangle?
Geometry
4-01 Exterior angles of a triangle
The vocabulary In this activity you will discover the relationship between an exterior angle and the sum of the
of geometry
two interior opposite angles. An exterior angle of a triangle is formed when one of the
sides is extended. B
Geometry
4-02 Step 1: Using your drawing program,
Angle sum of a copy the diagram on the right.
triangle Exterior angle
∠BCD
Geometry
4-03
Exterior angle A C D
of a triangle
108 N E W C E N T U R Y M A T H S 9 : S T A G E S 5.2/ 5.3
04_NC_Maths_9_Stages_5.2/5.3 Page 109 Friday, February 6, 2004 2:12 PM
Example 1
Find the value of the pronumeral in each of the following.
a b P
k°
52°
55°
42°
m°
Q R
Solution
a k + 55 + 42 = 180 (angle sum of a b ∠R = ∠Q = m° (∆PQR is isosceles)
triangle) ∴ m + m + 52 = 180 (angle sum of a
∴ k + 97 = 180 triangle)
∴ k = 83 ∴ 2m + 52 = 180
∴ 2m = 128
∴ m = 64
Example 2
Find the value of k in each of the following diagrams.
a b
59° 50°
k°
110° k°
42°
Solution
a k = 42 + 59 = 101 (exterior angle of triangle)
b k + 50 = 110
∴ k = 60 (exterior angle of triangle) E C
Example 3 75° B 55°
A
Find the size of ∠BDF.
Solution
∠DBC = ∠ABE = 75° (vertically opposite angles)
∴ ∠BDF = 75° + 55° (exterior angle of ∆BCD) D
= 130° F
Exercise 4-01
Example 1 1 Find the value of the pronumerals in the following diagrams.
SkillBuilder a b c 56°
23-03 40°
Angles in an
equilateral k°
triangle
d° 55°
SkillBuilder y°
23-04
Angles in a
scalene triangle
d e f
k°
w° 50°
k°
k°
a b m° 125° c
60° m°
55° 40°
m°
64°
132°
d e f
m°
m°
50°
m°
g h i 142°
m°
m°
44°
110° 21°
m°
a D E b c 53°
D C
80°
60° E
C
50°
B
45° F A
A B B F 60°
C
F
E C
d A e B f E
110° C
D 34° C
F
A E F
120°
F
B B
4 a One angle of an isosceles triangle is equal to 80°. Which of the following gives the sizes of
the other angles?
A 80° and 20° B 40° and 60° C 40° and 40° D 50° and 50°
b The largest angle in an isosceles triangle is equal to 70°. Which of the following gives the
sizes of the other angles?
A 70° and 55° B 110° and 70° C 70° and 40° D 55° and 55°
c One angle of an obtuse-angled triangle is equal to 50°. Which of the following gives the
sizes of the other angles?
A 80° and 50° B 100° and 30° C 65° and 65° D 60° and 70°
5 Find the values of the pronumerals.
a b c 40°
4a°
56°
3a° g°
2a°
4d°
60° 113°
d e f
y°
40°
g h i
40° m° 58°
n° 50°
w°
r°
135° 108° 3t° v°
2h° h°
j k h° l 58°
w°
2h° 2m°
6 Find the size of ∠XYZ in each of the following, giving reasons for each step.
a M b X c A
Y
Z
X X
75° 55°
105° B Y
N A
B
Y
Z Z C
A W
d e f
130° T
X 72°
B
Z X
U 30°
X Z
Y
Z Y
Y
7 a An isosceles triangle has a side length of 6 cm and one of its angles equal to 40°. Draw the
possible shapes of the triangle.
b The diagram below shows the shape of a roof truss. Calculate the sizes of the angles
indicated by the pronumerals.
50°
c°
a° b°
d°
8 a In ∆PQR, ∠Q is twice the size of ∠P, and ∠R is three times the size of ∠P. Find the size of
each angle.
b ∠P is 6° more than ∠Q, and ∠R is equal to the sum of ∠P and ∠Q. Find the measure of
each angle in ∆PQR.
9 In a triangle, one angle is three times the size of one of the other angles. What is the size of
each angle? (Assume that the angles are in whole degrees.) Is there more than one solution?
Explain.
68°
70°
40°
Properties of quadrilaterals
A quadrilateral is a four-sided plane figure. A quadrilateral may be either convex or non-
convex (also called concave).
A C
B
In the non-convex quadrilateral VWXY, the diagonal VX does V
not lie inside the figure.
W
X
Also, a convex quadrilateral has no interior angle greater than 180°.
T N
R
P
M
Q K
L
The interior angles P, Q, R and T of the In the non-convex quadrilateral KLMN, the
convex quadrilateral PQRT are all less interior angle M is greater than 180° (a
than 180°. reflex angle).
I NV E S T I GAT I NG GE OM E T R Y 113 CHAPTER 4
04_NC_Maths_9_Stages_5.2/5.3 Page 114 Friday, February 6, 2004 2:12 PM
Summary
The properties of the various types of quadrilaterals can be summarised as follows:
Trapezium – one pair of opposite sides parallel
Proof: R
T c°
Consider any quadrilateral in which the angles are a°, b°, c° d°
and d°.
a° b°
P Q
Draw in one diagonal PR as shown. R
Since ∠TPQ = ∠TPR + ∠QPR, T
d°
y° x°
a = p + q.
Similarly, c = x + y.
The quadrilateral has been divided into two triangles, each p°
having an angle sum of 180°. q° b°
P Q
∴ p + y + d = 180
q + b + x = 180
∴ (p + q) + b + (x + y) + d = 360
∴ a + b + c + d = 360
So the angle sum of a quadrilateral is 360°.
Using technology
Angle sum of a quadrilateral
Step 1: Construct a quadrilateral using your drawing program. Geometry
4-04
Step 2: Measure the four angles. Angle sum of a
Step 3: Add the four angles. What is the sum? quadrilateral
Step 4: Move (that is, drag) one of the vertices. What is the sum of the four angles of the new
quadrilateral?
Step 5: Construct one more quadrilateral and repeat steps 2–4. What can you say about the
angle sum of a quadrilateral?
Example 4
Find the value of m.
125° 93°
Solution
m + 78 + 125 + 93 = 360 (angle sum of a quadrilateral)
∴ m + 296 = 360 78°
∴ m = 64 m°
Example 5
B C
Find the value of d. d°
Solution 75°
∠CDA = 105° (angles on a straight line) A D E
∴ ∠ABC = 105° (opposite angles of a parallelogram are equal)
∴ d° = 105°
E
Example 6
Find the size of ∠BED. D
85°
70° 110°
Solution A B C
∠ABE = 70° (∆ABE is isosceles)
∴ ∠EBC = 180° − 70° = 110° (angles on a straight line))
∴ ∠BED + 110° + 110° + 85° = 360° (angle sum of quadrilateral BCDE)
∴ ∠BED = 55°
Note: There is another method of solving Example 9. Find this method.
Example 7
A rectangle is a parallelogram with one angle a right angle. Prove that each angle of a rectangle
is a right angle.
Solution
Consider the rectangle ABCD, with ∠A = 90°. D C
Since a rectangle is a parallelogram, AB DC and AD BC.
∴ ∠A + ∠D = 180° (co-interior angles, AB DC)
∴ ∠D = 180° − ∠A
= 180° − 90°
= 90° A B
∠C + ∠D = 180° (co-interior angles, AD BC)
∴ ∠C = 180° − ∠D
= 90°
∴ ∠B = 90° (angle sum of a quadrilateral)
∴ each angle of a rectangle is a right angle.
2 Refer to the table you constructed in Question 1, and name all quadrilaterals that have:
a no axes of symmetry b one pair of parallel sides Worksheet
4-05
c four equal sides d equal diagonals
Classifying
e opposite sides equal f four axes of symmetry quadrilaterals
g adjacent sides equal h one axis of symmetry
i opposite sides parallel j all angles measuring 90°
k two axes of symmetry l diagonals that bisect each other
m opposite angles equal n diagonals that meet at 90°
3 Which of the following statements are always true?
A A rhombus is a parallelogram.
B The diagonals of a parallelogram meet at right angles.
C A square is a rhombus.
D A parallelogram is a quadrilateral with a pair of opposite sides equal and parallel.
E A square is a rectangle.
F The diagonals of an isosceles trapezium bisect each other.
G The opposite angles of a rhombus are equal.
H The diagonals of a rhombus are equal and bisect each other at right angles.
m° 69° 110°
m°
25°
d e 100° f 95°
110°
123° (m + 20)°
2m° 3m°
83° m°
m°
Example 5 5 Find the value of the pronumeral in each of the following. (Give reasons for your answers.)
a b c
k° 125°
70° p°
a°
d e f 67°
68°
n°
35°
w°
t°
i
g h
70°
r° 110°
2a°
5c°
6 Find the size of ∠PQR in each of the following. (Give reasons for each step of your working.) Example 6
a b T c Q
R W Q
72° 80° R 74° T
110°
P V U P 30°
Q
W P R
d e P f T
Q
P S Q
60°
55°
35°
R 105° R
Q T 110°
R
C D P
7 Find the value of each pronumeral and give reasons.
a b q° c w°
58° 34°
p° v°
n°
t°
25° 112°
m°
d 2g° e y° f
a°
b° 68°
3g° k° c°
85°
A a B
Working mathematically
Reasoning: Too much or not enough?
1 State what information is required and what information is unnecessary in order to find
the value of t. Compare your answers with those of other students.
a 65° b c 123°
t° 70°
115° 68°
t°
t°
80° 70°
d 50° e f
t°
t°
40° 85° t°
60°
Properties of polygons
The general name for any plane figure
bounded by straight sides is a polygon.
Working mathematically
Reasoning and reflecting: Polygons
Work in groups of two or three people to complete these questions.
1 Triangles, parallelograms and kites are all polygons. Name three other polygons you have
encountered.
Convex polygons
A polygon may be either convex or non-convex. The diagonals of a convex polygon lie inside the
polygon. A convex polygon also has no interior angles greater than 180°.
convex polygon
D non-convex polygon
C
E
B
A
1 a Draw a hexagon (a six-sided polygon) and from one vertex draw all the diagonals.
b How many diagonals are there?
c How many triangles did you form?
d Using the method in the pentagon example above, calculate the sum of the interior
angles of a hexagon.
2 a Repeat the procedure in Question 1 for an octagon (an eight-sided polygon) and a
decagon (a 10-sided polygon). For each polygon, find the sum of its interior angles.
Number of Number of
Polygon Angle sum
sides triangles
triangle 3 1 180°
quadrilateral 4 2
pentagon 5
hexagon
octagon
decagon
c Copy and complete: The number of triangles formed is always two … than the number
of … of the polygon.
3 a What is the pattern for finding the angle sum of a polygon?
b What is the angle sum of a polygon with 20 sides?
c For a polygon of n sides, write a rule (a formula) for the sum
of the interior angles. Discuss your result with other students.
4 a Draw a non-convex polygon with 8 sides.
b Divide the polygon into triangles as shown.
c How many triangles have been formed?
d Calculate the sum of the interior angles of the eight-sided
non-convex polygon.
e Does your rule from Question 3c apply to the non-convex
polygon?
Example 8
Find the angle sum of a polygon with 18 sides.
Solution
Angle sum = (18 − 2) × 180°
= 16 × 180°
= 2880°
Example 9
Find the number of sides of a polygon that has an angle sum of 720°.
Solution
720 = (n − 2) × 180 or 180(n − 2) = 720
180n − 360 = 720
n − 2 = 720
---------
180 180n = 1080
=4
n = 1080
------------
∴n=4+2 180
=6 =6
∴ number of sides is 6.
Exercise 4-03
1 Use the rule above to find the angle sum of a polygon with: Example 9
Regular polygons
A regular polygon has all angles equal and all sides equal. Geometry
4-05
Polygons
The size of each angle of a regular polygon with n sides is then given by the following formula.
angle sum
One angle = ------------------------ , where n is the number of sides.
n
180 ( n – 2 )
= --------------------------
n
Example 10
Find the size of an angle of a six-sided regular polygon.
Solution
180 ( n – 2 )
Angle sum = (6 − 2) × 180° or Each angle = --------------------------, where n=6
n
= 4 × 180° = 180 × ( 6 – 2 )
--------------------------------
= 720° 6
∴ each angle = 720° ÷ 6 = 120°
= 120°
Example 11
The sum of the angles of a regular polygon is 6840°.
a How many sides does the polygon have?
b Find the size of each angle.
Solutions
a 6840 ÷ 180 = 38 or 180(n − 2) = 6840 (using angle sum formula)
∴ number of sides = 38 + 2 180n − 360 = 6840
= 40 180n = 7200
n= 7200
------------
180
= 40
b Each angle = 6840 ÷ 40
= 171°
Exercise 4-04
1 Triangles and quadrilaterals are polygons. What name is given to a regular:
a triangle?
b quadrilateral?
Example 10 2 Find the size of an interior angle of a regular:
a octagon b decagon
SkillBuilder c dodecagon (12 sides) d 15-sided polygon.
23-06
3 Find the size of the interior angle of a regular polygon that has:
Regular
polygons a 20 sides b 15 sides c 26 sides
d six sides e 30 sides f 10 sides.
Example 11 4 The sum of the interior angles of a regular polygon is 3600°.
a How many sides does the polygon have?
b Find the size of each interior angle (to one decimal place).
5 Find the size of an interior angle of a regular polygon whose angle sum is:
a 2520° b 3240° c 3060°
d 3960° e 1800° f 6120°
6 How many sides does a regular polygon have if the size of the interior angle is:
a 165°? b 170°?
Proof:
At each vertex, the sum of the interior angle and
exterior angle is 180°. So for an n-sided polygon,
the sum of all interior and exterior angles is 180n°. exterior angle
interior angle
But the sum of the interior angles is (n − 2) × 180° or
180(n − 2)°.
∴ Sum of exterior angles = sum of all interior angles and exterior angles − sum of interior angles
= 180n° − 180(n − 2)° Geometry
= 180n° − 180n° + 360° 4-06
= 360° Exterior angles
of a polygon
Using technology
Exterior angle sum of a convex polygon
Step 1: Use your drawing program to copy the diagram on the right.
Step 2: Measure the exterior angles shown and calculate their sum.
Step 3: Repeat steps 1 and 2 for a quadrilateral, pentagon and
hexagon. What is the sum of the exterior angles of a
convex polygon?
Example 12
The exterior angle of a regular polygon is 12°. How many sides does the polygon have?
Solution
Let n = the number of sides of the polygon.
∴ Sum of exterior angles = n × 12 = 360°
∴ 12n = 360
n = 360
---------
12
= 30
∴ the polygon has 30 sides.
Example 13
Find the size of each exterior angle of a regular 18-sided polygon.
Solution
Each exterior angle = 360°
----------- , where n is the number of sides.
n
Example 14
Each interior angle of a regular polygon is 160°. How many sides does the polygon have?
Solution
180 ( n – 2 ) °
Exterior angle = 180° − 160° (Why?) or Each angle = --------------------------
n
= 20° 180 ( n – 2 )
∴ -------------------------- = 160
Sum of exterior angles = 360° n
∴ number of exterior angles = 360 ÷ 20 = 18 180(n − 2) = 160n
∴ the polygon has 18 sides. 180n − 360 = 160n
180n = 160n + 360
20n = 360
n = 18
∴ the polygon has 18 sides.
Example 15
Find the size of each interior angle of a regular dodecagon (12 sides).
Solution
Each exterior angle = 360
---------
12
= 30°
∴ each interior angle = 180° − 30°
= 150°
Exercise 4-05
1 Find the number of sides of a regular polygon that has exterior angles equal to: Example 12
Power plus
1 Calculate the smaller angle between the clock hands at:
a 1:30pm b 12:15am c 4:45pm
2 a Construct ∆ABC with ∠A = 42°, ∠B = 67° and AB = 5.6 cm.
b Construct the largest possible square that is contained by a circle of radius 50 mm. What is
the length of the side of the square?
3 The angle sum of a polygon with n sides can also be given by the result (2n − 4) right angles.
Use this result to find the angle sum of a polygon with:
a seven sides b 16 sides.
4 The sum of the exterior angles of a polygon is 360°. Show C
that this result is true for the non-convex polygon ABCDE. B
E
D
5 a Draw polygons with 3, 4, 5, …, 10 sides. They do not need to be regular polygons.
b For each polygon, draw in all the diagonals and then copy and complete the table below.
Number of sides 3 4 5 … 10
Number of diagonals
c Find a formula that gives the number of diagonals, d, for a polygon of n sides.
Language of maths
alternate angle bisect closed
Worksheet co-interior complementary convex corresponding
4-08 degree dimension elevation interval
Geometry
crossword midpoint non-convex parallel pentagon
perpendicular polygon quadrilateral reflex
regular supplementary transversal vertex
1 Write four dictionary meanings of the word supplement (include the mathematical
meaning). How are the meanings related?
2 The word degree can be used in at least eight different ways. Write three sentences
using the word in different ways.
3 In the word polygon, what does ‘poly’ mean?
4 A hexagon is a six-sided polygon. What are the names of polygons with:
a 8 sides? b 5 sides? c 7 sides? d 9 sides?
5 A regular hexagon has 6 equal sides. What is the name of a regular polygon with:
a 3 sides? b 4 sides?
Topic overview
• Copy and complete:
Worksheet The best part of this chapter was …
4-09 The worst part was …
Geometry
summary New work included …
poster I need help with …
• Copy and complete the topic overview that has been started for you below. Remember to use
bright colours when completing your summary.
Add extra words and use pictures to make your overview useful to you. Have your overview
checked by other students or by your teacher to make sure nothing is missing or incorrect.
Quadrilaterals
• angle sum = 360°
•
INVESTIGATING
GEOMETRY Polygons
Triangles •
• acute, obtuse, right •
• •
• •
•
•
a b 3x° 72° c
52°
35°
127°
e°
t°
d e f 55°
3h°
2x° 67°
d°
98° 107°
c°
4a°
57° 108°
2 State whether or not the following statements are always true. Ex 4-01
a An equilateral triangle is an isosceles triangle.
b An isosceles triangle is a scalene triangle.
3 a Which types of quadrilateral have a pair of opposite sides parallel? Ex 4-02
b Which types of quadrilateral have two equal diagonals?
c Which types of quadrilateral have diagonals bisecting each other at right angles?
d Which types of quadrilateral have opposite sides equal?
e In which types of quadrilateral do the diagonals bisect the vertex angles through which
they pass?
f Which types of quadrilateral have two adjacent sides equal?
4 State whether the following statements are true or false. Ex 4-02
a A square is a parallelogram. b A rectangle is a square.
c A rectangle is a trapezium. d A parallelogram is a trapezium.
5 Find the value of each pronumeral in the following diagrams. Ex 4-02
a x° b m°
2x°
y° 2m°
70°
95° k° a°
c d
120° 150° 70°
g°
80°
6 Show that the angle sum of a decagon is 1440°. (A decagon has 10 sides.) Ex 4-03
7 Find the size (correct to one decimal place) of the interior angle of a regular polygon that Ex 4-04
has 13 sides.
8 The interior angles of a regular polygon are each 165°. How many sides does it have? Ex 4-05