Download Plane figure geometry

2
Plane figure geometry
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Turning
Measuring angles
Classifying and naming angles
Complementary and supplementary angles
Angles in a revolution
Bisecting angles
Constructing 90° angles to a line
Plane shapes
Points and lines
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Contents:
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34
PLANE FIGURE GEOMETRY
(Chapter 2)
If we look carefully we notice angles in many objects and situations.
For example, we see angles in building frameworks, pitches in roof structures, steepness of
ramps, and the positioning of boats and aeroplanes from their home bases.
Angle measurement dates back more than 2500 years and is still very important today in
architecture, building, surveying, engineering, navigation, space research, etc.
DEGREE MEASURE
To measure the size of angles accurately we use a small unit of measure. This unit is called
a degree.
There are 360 degrees, 360o , in one complete turn or revolution. Degrees have been used
from ancient Babylonian times when all the numbers were written and calculated in base 60.
360o is a useful measure because it is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, etc., and this
makes it easy to find fractions of 360o such as halves, thirds, quarters, and so on.
OPENING PROBLEM
There are four posts at the corners A,
B, C and D of a paddock with a raised
mound as shown. The distances
between the posts are easily measured
and are shown on the figure.
125 m
B
A
97 m
113 m
Your task is to draw an accurate scale diagram of
the situation. Because of the mound you cannot
accurately measure the distances AC or BD.
C
84 m
rough figure
D
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Things to discuss, think about and do:
² Is knowing the lengths of the four sides enough to accurately redraw the figure? Try
doing this using millimetres instead of metres.
² What other facts are necessary to accurately redraw the figure?
² How would you accurately measure the four angles at A, B, C and D in the real world?
² How would you accurately measure the four angles of the figure on paper?
² If the angle at A is known to be 100o , could you accurately peg out on the school oval
1
th scale model of paddock ABCD? You may only use four pegs, a large measuring
a 10
tape (as used on athletics days) and an angle measuring device (such as a theodolite or
modern laser equipment).
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PLANE FIGURE GEOMETRY
A
(Chapter 2)
35
TURNING
Many machines are operated with a dial showing different settings. The dial is rotated in
order for the machine to perform a particular operation.
Example 1
Self Tutor
Here is the dial of a ceiling fan.
There are three settings as well as
an ‘off’ position, as shown.
i
1
4
3
4
c
clockwise
What setting will be beside
the arrow for each of the
following rotations?
Through how many degrees has the dial turned in each case?
ii
a
anticlockwise
turn clockwise
b
1
2
turn anticlockwise
d
a full revolution anticlockwise
turn clockwise
a
i
low ii
1
4
of 360o = 90o
b
i
medium ii
c
i
low ii
3
4
of 360o = 270o
d
i
off ii
1
2
of 360o = 180o
360o
EXERCISE 2A
1 Using the fact that a full rotation or turn is equal to 360o , what is the size of:
a a half rotation
b a quarter turn
c a three-quarter turn
d three full turns
e one and a half rotations
f one-eighth rotation?
2 For the ceiling fan described in the example above, copy each diagram and fill in the
missing settings:
a
b
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3 An industrial fan is used to remove chemical fumes from a factory. It has five settings
as shown, and can only be turned clockwise.
OFF
clockwise
a There are six equal divisions on the dial.
How many degrees is each division worth?
VERY
VERY
HIGH
LOW
b How many degrees does each of the following
rotations require?
i off to medium
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36
PLANE FIGURE GEOMETRY
(Chapter 2)
B
MEASURING ANGLES
An angle is made up of two arms which meet at
a point called the vertex.
arm
The size of the angle is measured by using the
amount of turning (rotation) from one arm to the
other.
vertex
arm
angle
MEASURING DEVICES
We use a protractor (or geoliner)
which has tiny 1o markings on it.
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50 40
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centre
To use a protractor to measure angles we:
² place it so its centre is at the angle’s vertex and 0o lies
exactly on one arm
² start at 0o and follow the direction the angle turns through
to reach the other arm.
base line
The small °
symbol means
degrees.
Example 2
Self Tutor
Measure these angles:
a
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b This angle has size 123o :
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a This angle has size 47o .
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A
B
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PLANE FIGURE GEOMETRY
(Chapter 2)
37
EXERCISE 2B
1 What is the size of the angle being measured?
a
b
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38
PLANE FIGURE GEOMETRY
(Chapter 2)
2 Look at the following angles and estimate their sizes.
Record your estimates. Now measure each angle
accurately with a protractor and record your answers.
Copy and complete a table like the one given. How close
was your estimate to the measured angle?
Estimate
Measure
a
b
..
.
c
b
a
d
e
C
CLASSIFYING AND NAMING ANGLES
Angles are classified according to their size.
Revolution
Straight Angle
Right Angle
1
2
1
4
One complete turn.
One revolution = 360o :
turn.
1 straight angle = 180o :
turn.
1 right angle = 90o :
Acute Angle
Obtuse Angle
Reflex Angle
Less than a 14 turn.
An acute angle has size
between 0o and 90o :
Between 14 turn and 12 turn.
An obtuse angle has size
between 90o and 180o :
Between 12 turn and 1 turn.
A reflex angle has size
between 180o and 360o :
NAMING ANGLES
To avoid confusion in complicated figures, three point angle notation is used.
The shaded angle is named ]ADB or ]BDA
where the vertex must be the middle letter.
A
B
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Calling the shaded angle ]D, would be
unsatisfactory. Why?
5
D
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PLANE FIGURE GEOMETRY
39
(Chapter 2)
EXERCISE 2C
1 Match the names to the correct angles:
a ]ABC
b ]CAB
A
B
C
C
B
d
D
A
]CBD
B
B
A
B
]BCA
c
C
C
C
D
A
2 Draw and label a diagram for each of the following angles:
a ]DEF
b ]ZXY
c ]XYZ
d ]PQR
3 Using only a ruler and pencil, draw angles you estimate to be:
a 90o
b 45o
c 30o
d 60o
Check your estimations using a protractor.
4 Find the size of each of the following
angles without your protractor:
a ]ABC
b ]DBC
c ]ABD
d ]ABE
B
e
]RPQ
e
135o
A
35°
35° 24°
C
D
E
5 Use your ruler and protractor to draw angles with the following sizes:
a 35o
b 131o
c 258o
Get someone else to check the accuracy of your angles.
Z
6 Which is the larger angle,
]ABC or ]XYZ?
C
B
Y
A
X
7 Draw a free-hand sketch of:
a acute angle BPQ
b
d obtuse angle CPT
e
c
f
right angle NXZ
reflex angle DSM
straight angle QDT
revolution ]E
8 Use a protractor to measure the named angles:
a
i ]PMN ii ]OPL
b
i ]VTU ii reflex ]VST
iii ]PON
iii reflex ]TVU
S
Q
P
M
L
T
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40
PLANE FIGURE GEOMETRY
(Chapter 2)
9 Kim hits the billiard ball so
that it follows the path
shown.
What acute angle will it
make with the edge of the
table?
10
A golfer completing his swing holds the
golfclub behind his body. What is the size of
the reflex angle between his body and the club?
0
4
1
3
2
7
5
11 A fan switch has seven operating speeds apart
from the off position (0).
a If the knob is turned clockwise to 5,
through how many degrees has it rotated?
b How many less degrees would it have to turn if it rotated anticlockwise?
6
12 a
How many obtuse angles can be drawn
joining any three dots?
How many right angles can be drawn?
b
D
COMPLEMENTARY AND
SUPPLEMENTARY ANGLES
Two angles are complementary if their sizes add to 90o .
Two angles are supplementary if their sizes add to 180o .
For example,
O
U
M
P
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]STU and ]UTV are supplementary
because 79o + 101o = 180o .
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]MNO and ]MNP are complementary
because 52o + 38o = 90o .
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79° 101°
T
S
75
N
52°
38°
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PLANE FIGURE GEOMETRY
(Chapter 2)
41
Note: Two angles are equal if they have the same size (degree measure).
Example 3
Self Tutor
a Are angles with sizes 37o and 53o complementary?
b What angle size is the supplement of 48o ?
a 37o + 53o = 90o . So, the angles are complementary.
b The angle size is 180o ¡ 48o = 132o .
EXERCISE 2D
1 Add the following pairs of angles and state whether they are complementary,
supplementary, or neither:
b 30o , 150o
c 110o , 40o
a 20o , 70o
d 47o , 43o
e 107o , 63o
f 35o , 55o
2 Find the size of the angle complementary to:
a
30o
b
5o
c
85o
3 Find the size of the angle supplementary to:
a
100o
b
5o
c
90o
4 Classify the following angle pairs as complementary, supplementary or neither:
a ]BOC and ]COD
B
C
D
b ]AOC and ]COE
The symbol
c ]COD and ]DOE
indicates a
right
angle
d ]AOB and ]BOE
A
E
O
5 Copy and complete:
a the size of the complement of xo is ..........
b the size of the supplement of y o is ..........
Example 4
Self Tutor
Find the value of the pronumeral in:
a
b
x°
x°
58°
72°
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b The three angles add up to 180o .
) x = 180 ¡ 72 ¡ 78
) x = 30
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a The angles 58o and xo are
.
complementary
) x = 90 ¡ 58
) x = 32
78°
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42
PLANE FIGURE GEOMETRY
(Chapter 2)
6 Find the value of the pronumeral in:
a
b
c
a°
25°
b°
147°
d
c°
c°
e
62°
f
12°
f° f°
d°
e°
30°
33°
g
38°
h
i
32°
g°
g°
48°
h°
E
h°
x°
x°
74°
x°
ANGLES IN A REVOLUTION
A
The angles around a point total 360o .
D
So, ]ABC + ]CBD + ]DBA = 360o .
B
C
Example 5
Self Tutor
The sum of the five angles is 360o .
Find a in:
) the three equal angles add to
360o ¡ 90o ¡ 120o = 150o
a°
a°
120° a°
So, each must be 150o ¥ 3 = 50o
) a = 50:
EXERCISE 2E
1 Find the size of the angles marked with a pronumeral:
a
b
c
266°
z°
y°
27°
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PLANE FIGURE GEOMETRY
2 Find the value of the pronumerals in:
a
b
c
35°
109° 121°
p°
r°
60°
60° r°
60°
113°
q°
d
43
(Chapter 2)
e
f
138°
t° 41°
63°
t° 58°
52°
s°
F
u°
74°
u°
w°
BISECTING ANGLES
When we bisect an angle with a straight line we divide it into two angles of equal size.
The following construction shows angle bisection with a compass and ruler only.
Example 6
Self Tutor
A
Bisect angle ABC.
B
Step 1:
C
A
With centre B, draw an arc of convenient
radius which cuts BA and BC at P and Q
respectively.
P
B
C
Q
A
Step 2:
With Q as centre, draw an arc within the
angle ABC.
P
B
C
Q
A
Step 3:
Keeping the same radius and with centre P
draw another arc to intersect the previous
one at M.
P
M
B
C
Q
A
Step 4:
Join B to M. BM bisects angle ABC,
so ]ABM = ]CBM.
P
M
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44
PLANE FIGURE GEOMETRY
(Chapter 2)
EXERCISE 2F
1 Use your protractor to draw an angle ABC of size 80o :
a Bisect angle ABC, using a compass and ruler
only.
b Check with your protractor the size of each of the
two angles you constructed.
2 Draw an acute angle XYZ of your own choice.
a Bisect the angle without using a protractor.
b Check your construction using your protractor.
3 Draw an obtuse angle ABC of your own choice.
a Bisect the angle using a compass only.
b Check your construction using your protractor.
a Using a ruler, draw any triangle with sides greater than 5 cm.
b Bisect each angle using a compass and straight edge only.
c What do you notice about the three angle bisectors?
4
G CONSTRUCTING 90° ANGLES TO A LINE
A right angle or 90o angle can be constructed without a protractor or set square. Consider
the following example:
Example 7
Self Tutor
Construct an angle of 90o at P on the line
segment XY.
Step 1:
Step 2:
X
On a line segment XY, draw a semicircle with centre P and convenient
radius which cuts XY at M and N.
X
With centre M and convenient radius
larger than MP, draw an arc above P.
X
Y
P
M
P
N
M
P
N
Y
Y
W
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With centre N and same radius draw
an arc to cut the first one at W.
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Step 3:
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PLANE FIGURE GEOMETRY
Step 4:
Draw the line from P through W.
Angles WPY and WPX are 90o .
45
(Chapter 2)
W
X
M
N
P
Y
EXERCISE 2G
1
a Draw a horizontal line segment AB which is 5 cm long.
b Use your ruler to find C on AB where AC = 2:5 cm.
c Construct a 90o angle at C using your compass and ruler.
2
a Draw a line segment AB of length 3 cm.
b Construct an angle of 90o at A. (You will need to extend
the line segment AB on the other side of A.)
c Draw AC of length 4 cm, such that ]BAC is 90o .
d Join BC to form triangle ABC and measure the length of
BC.
3
a
b
c
d
e
f
Draw a line segment XY of length 8 cm.
At X construct an angle of 90o .
Draw XZ of length 5 cm. Join ZY.
Use your compass to bisect angle ZXY.
What is the size of angle YXP?
If ZY and XP meet at Q, measure QY to
the nearest mm.
Horizontal
lines are drawn
across the page
like a horizon.
P
Z
Q
5 cm
X
H
8 cm
Y
PLANE SHAPES
A shape that is drawn on a flat surface or plane is
called a plane figure.
If the shape has no beginning or end it is said to be
closed.
We can use properties like these to classify objects.
POLYGONS
Polygons are closed plane figures with straight
line sides which do not cross over themselves.
Simple examples are:
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quadrilaterals
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triangles
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Y:\HAESE\SA_08-6ed\SA08-6_02\045SA08-6_02.CDR Thursday, 26 October 2006 2:08:58 PM PETERDELL
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46
PLANE FIGURE GEOMETRY
(Chapter 2)
Example 8
Self Tutor
Classify as a polygon or not, giving reasons:
a
b
c
a It is closed and has straight line sides, so it is a polygon.
b It does not have all straight line sides, so it is not a polygon.
c
It is not closed and so it is not a polygon.
Polygons are named according to the number of
sides they have. For example, a 9 sided polygon
can be called a 9-gon. However, many polygons
are known by other more familiar names. Here
are the first few:
Number of Sides
3
4
5
6
7
8
9
10
Polygon Name
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
CIRCLES
Suppose we make a loop from a length of light rope and place it over a fixed spike in the
ground. The rope is made taut and a stick is placed at the opposite end to the fixed spike.
By keeping the rope taut and moving the stick around the spike, a circle is produced.
The fixed spike is the circle’s centre.
This method of drawing a circle was known
and used by builders in ancient Egypt.
Parts of a circle
this is a circle
this is the circle’s centre (and is often not drawn)
this is a radius which joins the circle’s centre to
any point on the circle
AB is a diameter of the circle as it passes through
the circle’s centre and its end points lie on the circle
A
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B
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Y:\HAESE\SA_08-6ed\SA08-6_02\046SA08-6_02.CDR Thursday, 26 October 2006 2:09:17 PM PETERDELL
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PLANE FIGURE GEOMETRY
(Chapter 2)
47
this is a semi-circle (half of a circle)
this is an arc of the circle (part of the circle)
this is a chord of the circle
this is a segment of the circle
this is a sector of the circle
EXERCISE 2H
1 Classify as polygons or not, giving reasons:
a
b
c
d
convex
(no reflex
angles)
e
f
g
h
non-convex
(has a reflex angle)
2 Draw a free-hand sketch of:
a a convex 4-sided polygon
c a convex 5-sided polygon
ACTIVITY
a non-convex 4-sided polygon
a non-convex 6-sided polygon
b
d
3 What special name is given to a polygon with:
a three sides
b four sides
d six sides
e eight sides
c
f
five sides
ten sides
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B
C
D
E
F
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H
I
J
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g
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4 Match the part of the figure indicated to the phrase which best describes it:
a
b
c
d
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Y:\HAESE\SA_08-6ed\SA08-6_02\047SA08-6_02.CDR Friday, 27 October 2006 3:06:07 PM PETERDELL
a
a
a
a
a
a
a
a
a
a
semi-circle
radius
minor arc
major arc
diameter
chord
minor segment
major segment
major sector
minor sector
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48
PLANE FIGURE GEOMETRY
(Chapter 2)
GROUP ACTIVITY
GEOMETRICAL IDEAS
What to do:
1 In the classroom find two examples of each of the following:
² a point
² a line
² an angle
² a flat surface
² a curved surface
2 Discuss the following:
² What is meant by a point?
I
² How small can a point be?
POINTS AND LINES
POINTS IN GEOMETRY
Examples of a point in the classroom are:
² the intersection of two adjacent walls and the floor
² a speck of dust in the room at a particular instant in time.
In geometry, a point could be represented by a small dot. To help identify it we could name
it with or assign a capital letter.
Consider:
The letters A, B and C are useful for identifying
points to which we are referring.
B
We can make statements like:
“the distance from A to B is .....” or
“the angle at B measures .....”, etc.
A
C
a point marks a position and does not have any size.
To a mathematician,
FIGURES AND VERTICES
A
Any special collection of points in a plane (flat
surface) is called a figure.
The figure alongside contains four points which
have been labelled A, B, C and D. These corner
points are also known as vertices.
B
C
D
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Vertices is the
plural of vertex.
(Note:
Point B is a vertex of the figure.)
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Y:\HAESE\SA_08-6ed\SA08-6_02\048SA08-6_02.CDR Thursday, 26 October 2006 2:09:34 PM PETERDELL
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PLANE FIGURE GEOMETRY
(Chapter 2)
GROUP DISCUSSION
49
LINES
What to do:
Discuss the following questions:
1 How many different straight lines could be drawn through a single point?
2 Suppose A and B are two different points. How many straight lines can
be drawn such that each line passes through both A and B?
STRAIGHT LINES
A straight line (usually called a line) is a continuous infinite collection of points
which lie in a particular direction.
B
This line which passes through points A and B is
called “line AB” or “line BA”.
A
A
This part line which joins points A and B is called
“line segment AB” or “line segment BA”.
B
Note:
²
When three or more points lie on a line we say
that the points are collinear. The points A, B, C
and D shown are collinear.
²
If three or more lines meet (intersect) at the same
point we say that the lines are concurrent. The
lines shown are concurrent at point B.
B
A
C D
B
PARALLEL AND INTERSECTING LINES
Imagine a plane, like a table top, which goes
on indefinitely in all directions, i.e., it has
no boundaries. In such a plane, two straight
lines are either parallel or intersecting. Arrow heads are used to show parallel lines.
point of
intersection
parallel
Parallel lines are lines which are always a fixed distance apart and so never meet.
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a point of intersection
concurrent lines
100
c
f
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2 Describe with a sketch the meaning of:
a a vertex
b an angle
d parallel lines
e collinear points
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a flat surface
25
c
0
1 Give two examples in the classroom of:
a a point
b a line
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EXERCISE 2I
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Y:\HAESE\SA_08-6ed\SA08-6_02\049SA08-6_02.CDR Thursday, 26 October 2006 2:09:46 PM PETERDELL
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50
PLANE FIGURE GEOMETRY
(Chapter 2)
3 Give all ways of naming the following lines:
a
b
C
B
Q
P
(6 answers)
A
4 Name the point of intersection of:
a line 1 and line 2
b line 2 and line 3
c line AB and line MN
M
A
line 3
line 2
line 1
B
C N
5 ABCD is a quadrilateral and line segment BD is called a diagonal.
a Name the four sides of the quadrilateral.
A
b Name the two diagonals of the quadrilateral.
c At what point do the diagonals meet?
d How many line segments meet at A?
e What can be said about points A, X and C?
f What can be said about the line segments AB,
D
DB and CB?
B
X
C
6 How many different lines do you think you can draw through:
a two points A and B
b all three collinear points A, B and C
c one point A
d all three non-collinear points A, B, C?
7 Draw a different diagram to fit each statement:
a C is a point on line AB.
b Lines AB and CD meet at point X.
c Point A does not lie on line BC.
d X, Y and Z are collinear.
e Line segments AB, CD and EF are concurrent at G.
a Name line AC in two other ways.
b Name two different lines containing point B.
c What can be said about
i points A, B and D
ii lines BF and AD
iii lines FE, CD and AB?
8
A
B
C
D
F
E
9 When drawing lines through three different points there are two possible cases:
3 different lines
1 line, as the points are collinear
a How many different cases can we have for four different points? Illustrate each
case.
b Now draw the cases for five different points.
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c You may also wish to experiment with a higher number of points.
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Y:\HAESE\SA_08-6ed\SA08-6_02\050SA08-6_02.CDR Thursday, 26 October 2006 2:09:53 PM PETERDELL
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PLANE FIGURE GEOMETRY
51
(Chapter 2)
REVIEW SET 2A
1
a State the complement of 41o .
b What is the size of one revolution?
c Determine the measure of angle ABC.
C
A
B
2 For the given figure:
a find the angle number corresponding to:
i ]BDA ii ]DCB iii ]BAC
b classify the following angles as acute,
obtuse or reflex:
i 3
ii 1 iii 4
A
1
119°
2
6
5
D
4
C
3 Find the value of the pronumeral in:
a
b
n°
3
B
c
x°
x°
68°
y° 100°
140°
4 Name the following polygons:
a
b
a
5 Define the following terms:
an arc of a circle
b
a sector of a circle
6 If the radius of a circle is 5 cm in length, what is the length of a diameter?
7 How many points are needed to determine the position of a line?
8 Draw a diagram to illustrate the following statement:
“Line segments AB and CD intersect at P.”
9
a
b
R
Q
S
U
P
Name line RS in two other ways.
Name two different lines containing
point P.
What can be said about
i points P, Q and R
ii lines PQ and RS?
c
T
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10 Use your protractor to draw an angle of 56o . Bisect this angle using your compass.
Check that the two angles produced are 28o each.
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Y:\HAESE\SA_08-6ed\SA08-6_02\051SA08-6_02.CDR Thursday, 26 October 2006 2:10:12 PM PETERDELL
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52
PLANE FIGURE GEOMETRY
(Chapter 2)
REVIEW SET 2B
1
a Draw a diagram to illustrate an obtuse angle.
b If angle ABC and angle DEF are equal angles
and the measure of angle ABC is 72o , find the
measure of angle DEF.
c Determine the measure of the reflex angle PQR.
P
Q
R
A
2 Match the angle name to the angle
number:
a ]ADC
b ]BAC
1 6
]ABD
c
2
B
3 4
C
3 Match each angle description to one of the angles shown:
a revolution
b right angle
c
d reflex angle
e acute angle
f
D
C
E
F
a What is the complement of 63o ?
4
D
straight angle
obtuse angle
B
A
5
b What is the supplement of 70o ?
5 Find, giving a reason, the value of y in each of the following:
a
b
c
y°
131°
y°
123°
y°
y°
y°
6 Name the following polygons: a
b
7 Draw a circle with radius 2 cm. Mark and label:
a a radius
b a sector
c a segment
d
a minor arc
8 Copy and complete: “Points are collinear if ......”.
9 State the total number of points of intersection
of the lines:
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10 Draw AB of length 4 cm. Construct an angle of 90o at B using a compass and ruler
only, and draw BC of length 3 cm and perpendicular to AB. Draw in and measure the
length of AC.
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Y:\HAESE\SA_08-6ed\SA08-6_02\052SA08-6_02.CDR Monday, 30 October 2006 9:08:43 AM PETERDELL
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