points, lines and surfaces · basic elements of geometry 15 a surface can be either flat or curved....
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Basic Elements of Geometry
Parallel Lines and Perpendicular Lines
1
Let’
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ave
a M
ath
Ch
at
Points, Lines and Surfaces
Basic Elements of Geometry
2
What is geometry?
Basic Elements of Geometry
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Geometry is an old branch of mathematics which
studies shapes and sizes of figures.
Points, lines and surfaces are the three basic
elements of geometry.
Basic Elements of Geometry
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What is a point?
Basic Elements of Geometry
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A point which has no size,
represents a position in space or on a plane.
Label a point by a capital letter.
Use a small cross ‘×’ or a small dot ‘’
to represent a point.
e.g. × A B
We call it ‘point A’. We call it ‘point B’.
Basic Elements of Geometry
6
What is a line?
Basic Elements of Geometry
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A line which has no width,
is made up of an infinite number of
points and can extend indefinitely.
Use a letter (e.g. L or l) to label a line.
Lines can either be straight line or curves.
e.g. L
l
straight line curve
Basic Elements of Geometry
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We can also take any two points on a
straight line and label the straight line
by those two points.
e.g.
M N × ×
line MN or line NM
Basic Elements of Geometry
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What is a line
segment?
Basic Elements of Geometry
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A line segment is a part of a straight line
and has two end-points.
e.g.
X Y × ×
line segment XY or
line segment YX
Label a line segment by its end-point.
Basic Elements of Geometry
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The length of a line segment is the
distance between the end-points.
Sometimes, we can use the name of the
line segment to denote its length.
e.g.
X Y × ×
The length of the line segment XY is 8 cm,
or we can simply write XY = 8 cm.
P R
T
Q
S
Basic Elements of Geometry
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In the figure, line segments PQ
and RS and drawn.
(a) Mark a point U on PQ and draw the line segment RU.
(b) Name the longest line segment in the figure.
U
PQ
P R
T
Q
S
Basic Elements of Geometry
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The point at which two straight lines intersect is
called the point of intersection.
Points of intersection can also be made by curves.
T is the point of
intersection of PQ and RS.
Basic Elements of Geometry
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What is a surface?
Basic Elements of Geometry
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A surface can be either flat or curved.
A flat surface is called a plane.
When a surface is curved, such kind of
surface is a curved surface.
e.g.
A plane has no thickness
and contains infinite number
of points and lines.
plane
curved
surface
A plane can extend endlessly.
We can label a plane by a
Greek letter π.
π π
Parallel Lines and Perpendicular Lines
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Now I am going to introduce parallel lines and
perpendicular lines.
A
B C
D
E
F
X Y
V Z
Parallel Lines and Perpendicular Lines
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Parallel lines are straight lines on the same plane
which never intersect each other no matter how
far they are extended.
e.g.
Use the symbol ‘//’ to represent ‘is parallel to’.
XY // VZ, XV // YZ
Same number of arrowheads (e.g. ‘’ or ‘ ’)
on lines in the same direction are used to
indicate that the lines are parallel.
Parallel Lines and Perpendicular Lines
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Perpendicular lines are two straight lines
which intersect and form a right angle.
e.g.
Use the symbol ‘’ to represent
‘is perpendicular to’.
The mark of a right angle ‘ ’ is used to indicate that the
two lines are perpendicular.
M
O
N
P
M
O
N
P
MN OP MN OP
Parallel Lines and Perpendicular Lines
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The distance between two parallel
lines is always the same.
e.g.
parallel
lines
distance
B
D
C
E
A
F
G
Parallel Lines and Perpendicular Lines
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AB // GF, BC // FE
(a) Mark parallel lines in the figure with arrowheads.
Example 1
(b) Write down all pairs of parallel lines in the figure.
Parallel Lines and Perpendicular Lines
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Example 2
Write down all pairs of perpendicular lines in the figure.
L1
L2
L3 L4 L5
L1 L3, L2 L4